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Tight span

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In <a href="Metric_geometry" class="mw-redirect" title="Metric geometry">metric geometry</a>, the metric envelope or tight span of a <a href="Metric_space" title="Metric space">metric space</a> M is an <a href="Injective_metric_space" title="Injective metric space">injective metric space</a> into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the <a href="Convex_hull" title="Convex hull">convex hull</a> of a point set in a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the <a href="Injective_hull" title="Injective hull">injective hull</a> of a <a href="Module_(mathematics)" title="Module (mathematics)">module</a> in <a href="Algebra" title="Algebra">algebra</a>, a concept with a similar description relative to the <a href="Category_(mathematics)" title="Category (mathematics)">category</a> of R-modules rather than metric spaces.

The tight span was first described by <a href="#CITEREFIsbell1964">Isbell (1964)</a>, and it was studied and applied by Holsztyński in the 1960s. It was later independently rediscovered by <a href="#CITEREFDress1984">Dress (1984)</a> and <a href="#CITEREFChrobakLarmore1994">Chrobak & Larmore (1994)</a>; see <a href="#CITEREFChepoi1997">Chepoi (1997)</a> for this history. The tight span is one of the central constructions of <a href="T-theory" title="T-theory">T-theory</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The tight span of a metric space can be defined as follows. Let (X,d) be a metric space, and let T(X) be the set of extremal functions on X, where we say an extremal function on X to mean a function f from X to R such that

In particular (taking x = y in property 1 above) f(x) ≥ 0 for all x. One way to interpret the first requirement above is that f defines a set of possible distances from some new point to the points in X that must satisfy the <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a> together with the distances in (X,d). The second requirement states that none of these distances can be reduced without violating the triangle inequality.

The tight span of (X,d) is the metric space (T(X),δ), where

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<annotation encoding="application/x-tex">{\displaystyle \delta =(\inf\{C\in \mathbb {R} _{\geq 0}:|g(x)-f(x)|\leq C{\text{ for all }}x\in X\})_{f,g\in T(X)}=(\|g-f\|_{\infty })_{f,g\in T(X)}}</annotation>

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is analogous to the metric induced by the <a href="Lp_space#General_ℓp-space" title="Lp space">ℓ∞ norm</a>. (If d is bounded, then δ is the subspace metric induced by the metric induced by the <a href="Lp_space#General_ℓp-space" title="Lp space">ℓ∞ norm</a>. If d is not bounded, then every extremal function on X is unbounded and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)\not \subseteq \ell ^{\infty }(X).}">

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<annotation encoding="application/x-tex">{\displaystyle T(X)\not \subseteq \ell ^{\infty }(X).}</annotation>

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</math><img src="./ec8f5c43636a40d073faf50f44bbd7260b01bee6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.805ex; height:3.176ex;" alt="{\displaystyle T(X)\not \subseteq \ell ^{\infty }(X).}" loading="lazy"> Regardless, it will be true that for any f,g in T(X), the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g-f}">

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<annotation encoding="application/x-tex">{\displaystyle g-f}</annotation>

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</math><img src="./bfadf2c2279e1c4e1ad815ca1346fe276cfeef64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle g-f}" loading="lazy"> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(X)}">

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(X)}</annotation>

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</math><img src="./f14f214b4c5420360eb91a5316fa1525f1b12ac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.634ex; height:2.843ex;" alt="{\displaystyle \ell ^{\infty }(X)}" loading="lazy">, i.e., is bounded.)

<div class="mw-heading mw-heading2"><h2 id="Equivalent_definitions_of_extremal_functions">Equivalent definitions of extremal functions</h2></div>

For a function f from X to R satisfying the first requirement, the following versions of the second requirement are equivalent:

<li>f is pointwise minimal with respect to the aforementioned first requirement, i.e., for any function g from X to R such that d(x,y) ≤ g(x) + g(y) for all x,y in X, if g≤f pointwise, then f=g.<a href="#cite_note-KK-2">[2]</a>: 93, Proposition 4.6.2 <a href="#cite_note-3">[Note 1]</a><a href="#cite_note-4">[Note 2]</a><a href="#cite_note-DHKMS-5">[3]</a>: Lemma 5.1 </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Basic_properties_and_examples">Basic properties and examples</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle 0\leq f(x).}</annotation>

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</math><img src="./04da36de1f7c8e7d82ef0a11c7bdbc8682ecdce1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.325ex; height:2.843ex;" alt="{\displaystyle 0\leq f(x).}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle (d(x,y))_{y\in X}}</annotation>

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</math><img src="./591839dcbf7c5bfd9ce2e45d531255b48c35c848.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.899ex; height:3.009ex;" alt="{\displaystyle (d(x,y))_{y\in X}}" loading="lazy"> is extremal. (Proof: Use symmetry and the <a href="Triangle_inequality#Metric_space" title="Triangle inequality">triangle inequality</a>.)<a href="#cite_note-6">[Note 3]</a></li>

<li>If X is finite, then for any function f from X to R that satisfies the first requirement, the second requirement is equivalent to the condition that for each x in X, there exists y in X such that f(x) + f(y) = d(x,y). (If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\emptyset ,}">

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<annotation encoding="application/x-tex">{\displaystyle X=\emptyset ,}</annotation>

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</math><img src="./672babc7833e2affe32bfbb54b94672d70d38ce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.888ex; height:2.676ex;" alt="{\displaystyle X=\emptyset ,}" loading="lazy"> then both conditions are true. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\neq \emptyset ,}">

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<annotation encoding="application/x-tex">{\displaystyle X\neq \emptyset ,}</annotation>

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</math><img src="./21d6677dde1ac34e391ff59b216eb593f1259055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.888ex; height:2.843ex;" alt="{\displaystyle X\neq \emptyset ,}" loading="lazy"> then the supremum is achieved, and the first requirement implies the equivalence.)</li>

<li>Say |X|=2, and choose distinct a, b such that X={a,b}. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)=\{f\in (\mathbb {R} _{\geq 0})^{X}:f(a)+f(b)=d(a,b)\}}">

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<annotation encoding="application/x-tex">{\displaystyle T(X)=\{f\in (\mathbb {R} _{\geq 0})^{X}:f(a)+f(b)=d(a,b)\}}</annotation>

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</math><img src="./48a1600b11cb87a84c5196db1acbb1ffcf772cdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.986ex; height:3.176ex;" alt="{\displaystyle T(X)=\{f\in (\mathbb {R} _{\geq 0})^{X}:f(a)+f(b)=d(a,b)\}}" loading="lazy"> is the convex hull of {{(a,1),(b,0)},{(a,0),(b,1)}}. [Add a picture. Caption: If X={0,1}, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)=\{v\in (\mathbb {R} _{\geq 0})^{2}:v_{0}+v_{1}=d(0,1)\}}">

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<annotation encoding="application/x-tex">{\displaystyle T(X)=\{v\in (\mathbb {R} _{\geq 0})^{2}:v_{0}+v_{1}=d(0,1)\}}</annotation>

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</math><img src="./6fb36cb7fbc0d98d16c1948f2cadffb087c7aed6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.315ex; height:3.176ex;" alt="{\displaystyle T(X)=\{v\in (\mathbb {R} _{\geq 0})^{2}:v_{0}+v_{1}=d(0,1)\}}" loading="lazy"> is the convex hull of {(0,1),(1,0)}.]<a href="#cite_note-HRS-7">[4]</a>: 124 </li>

<li>Every extremal function f on X is Katetov:<a href="#cite_note-8">[5]</a><a href="#cite_note-9">[6]</a>: Section 2 f satisfies the first requirement and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\in X\quad f(x)\leq d(x,y)+f(y),}">

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<annotation encoding="application/x-tex">{\displaystyle \forall x,y\in X\quad f(x)\leq d(x,y)+f(y),}</annotation>

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</math><img src="./9554a8529a04583dd86f77a207e5185df6456449.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.746ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in X\quad f(x)\leq d(x,y)+f(y),}" loading="lazy"> or equivalently, f satisfies the first requirement and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\in X\quad |f(y)-f(x)|\leq d(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle \forall x,y\in X\quad |f(y)-f(x)|\leq d(x,y)}</annotation>

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</math><img src="./89daab76c598643d384b8dfb9eed903c2a1c6958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.393ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in X\quad |f(y)-f(x)|\leq d(x,y)}" loading="lazy"> (is 1-<a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz</a>), or equivalently, f satisfies the first requirement and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in X\quad \sup\{f(y)-d(x,y):y\in X\}=f(x).}">

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<annotation encoding="application/x-tex">{\displaystyle \forall x\in X\quad \sup\{f(y)-d(x,y):y\in X\}=f(x).}</annotation>

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</math><img src="./d170f8036cd6e2159cdd8f789cba1f711d8e5ae5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.683ex; height:2.843ex;" alt="{\displaystyle \forall x\in X\quad \sup\{f(y)-d(x,y):y\in X\}=f(x).}" loading="lazy"><a href="#cite_note-KK-2">[2]</a>: Proof of Proposition 4.6.1 <a href="#cite_note-10">[Note 4]</a></li>

<li>T(X)⊆<a href="Continuous_functions_on_a_compact_Hausdorff_space" class="mw-redirect" title="Continuous functions on a compact Hausdorff space">C(X)</a>. (Lipschitz functions are continuous.)</li>

<li>T(X) is <a href="Equicontinuous" class="mw-redirect" title="Equicontinuous">equicontinuous</a>. (Follows from every extremal function on X being 1-Lipschitz; cf. <a href="Equicontinuity#Examples" title="Equicontinuity">Equicontinuity#Examples</a>.)</li>

<li>Not every Katetov function on X is extremal. For example, let a, b be distinct, let X = {a,b}, let d = ([x≠y])x,y in X be the <a href="Discrete_metric" class="mw-redirect" title="Discrete metric">discrete metric</a> on X, and let f = {(a,1),(b,2)}. Then f is Katetov but not extremal. (It is almost immediate that f is Katetov. f is not extremal because it fails the property in the third bullet of this section.)</li>

<li>If d is bounded, then every f in T(X) is bounded. In fact, for every f in T(X), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty }\leq \|d\|_{\infty }.}">

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<annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }\leq \|d\|_{\infty }.}</annotation>

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</math><img src="./9a826c8777359396a5c425ac9ee1b0fe20b4b281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.64ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }\leq \|d\|_{\infty }.}" loading="lazy"> (Note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\in \ell ^{\infty }(X\times X).}">

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<annotation encoding="application/x-tex">{\displaystyle d\in \ell ^{\infty }(X\times X).}</annotation>

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</math><img src="./6323c1a18f69ce0dc3ddcfefca18d2e6e314d586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.158ex; height:2.843ex;" alt="{\displaystyle d\in \ell ^{\infty }(X\times X).}" loading="lazy">) (Follows from the third equivalent property in the above section.)</li>

<li>If d is unbounded, then every f in T(X) is unbounded. (Follows from the first requirement.)</li>

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<annotation encoding="application/x-tex">{\displaystyle T(X)}</annotation>

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</math><img src="./fe67aad4eff628fcb5bb28ee6a2213d28ff12e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle T(X)}" loading="lazy"> is closed under pointwise limits. For any pointwise convergent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in (T(X))^{\omega },}">

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<annotation encoding="application/x-tex">{\displaystyle f\in (T(X))^{\omega },}</annotation>

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</math><img src="./432cfa745965cee84164ac2eba6c0cdcc5355a03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.256ex; height:2.843ex;" alt="{\displaystyle f\in (T(X))^{\omega },}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim f\in T(X).}">

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<annotation encoding="application/x-tex">{\displaystyle \lim f\in T(X).}</annotation>

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</math><img src="./7e28489e7aacc0ccd85add927875aaa446a47366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.808ex; height:2.843ex;" alt="{\displaystyle \lim f\in T(X).}" loading="lazy"></li>

<li>If (X,d) is compact, then (T(X),δ) is compact.<a href="#cite_note-11">[7]</a><a href="#cite_note-KK-2">[2]</a>: Proposition 4.6.3 (Proof: The <a href="Extreme_value_theorem#Generalization_to_metric_and_topological_spaces" title="Extreme value theorem">extreme-value theorem</a> implies that d, being continuous as a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X\to \mathbb {R} ,}">

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<annotation encoding="application/x-tex">{\displaystyle X\times X\to \mathbb {R} ,}</annotation>

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</math><img src="./5c6dddd42a8ec19611649c3c3ed3fbe257ccd202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.739ex; height:2.509ex;" alt="{\displaystyle X\times X\to \mathbb {R} ,}" loading="lazy"> is bounded, so (see previous bullet) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)\subseteq \{f\in C(X):\|f\|_{\infty }\leq \|d\|_{\infty }\}}">

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<annotation encoding="application/x-tex">{\displaystyle T(X)\subseteq \{f\in C(X):\|f\|_{\infty }\leq \|d\|_{\infty }\}}</annotation>

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</math><img src="./9d816c92cbc22e809b325cc2d213d969f2ddca43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.455ex; height:2.843ex;" alt="{\displaystyle T(X)\subseteq \{f\in C(X):\|f\|_{\infty }\leq \|d\|_{\infty }\}}" loading="lazy"> is a bounded subset of C(X). We have shown T(X) is equicontinuous, so the <a href="Arzel%C3%A0%E2%80%93Ascoli_theorem" title="Arzelà–Ascoli theorem">Arzelà–Ascoli theorem</a> implies that T(X) is <a href="Relatively_compact" class="mw-redirect" title="Relatively compact">relatively compact</a>. However, the previous bullet implies T(X) is closed under the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

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</math><img src="./8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"> norm, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation>

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</math><img src="./8348195cf09473662c6f59e6717722a6fc01d0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" loading="lazy"> convergence implies pointwise convergence. Thus T(X) is compact.)</li>

<li>For any function g from X to R that satisfies the first requirement, there exists f in T(X) such that f≤g pointwise.<a href="#cite_note-KK-2">[2]</a>: Lemma 4.4 </li>

<li>For any extremal function f on X, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in X\quad f(x)=\sup\{|f(y)-d(x,y)|:y\in X\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \forall x\in X\quad f(x)=\sup\{|f(y)-d(x,y)|:y\in X\}.}</annotation>

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</math><img src="./a5f9d6a92b1ffc50616c4297cf47739923e0e0eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.59ex; height:2.843ex;" alt="{\displaystyle \forall x\in X\quad f(x)=\sup\{|f(y)-d(x,y)|:y\in X\}.}" loading="lazy"><a href="#cite_note-KK-2">[2]</a>: Proposition 4.6.1 <a href="#cite_note-12">[Note 5]</a></li>

<li>For any f,g in T(X), the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g-f}">

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<annotation encoding="application/x-tex">{\displaystyle g-f}</annotation>

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</math><img src="./bfadf2c2279e1c4e1ad815ca1346fe276cfeef64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle g-f}" loading="lazy"> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(X)}">

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(X)}</annotation>

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</math><img src="./f14f214b4c5420360eb91a5316fa1525f1b12ac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.634ex; height:2.843ex;" alt="{\displaystyle \ell ^{\infty }(X)}" loading="lazy">, i.e., is bounded. (Use the above bullet.)</li>

<li>The Kuratowski map<a href="#cite_note-HRS-7">[4]</a>: 125 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e:=((d(x,y))_{y\in X})_{x\in X}}">

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<annotation encoding="application/x-tex">{\displaystyle e:=((d(x,y))_{y\in X})_{x\in X}}</annotation>

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</math><img src="./08bdc7696640590c2b22eb99596738a4eb0c53da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.206ex; height:3.009ex;" alt="{\displaystyle e:=((d(x,y))_{y\in X})_{x\in X}}" loading="lazy"> is an <a href="Isometry" title="Isometry">isometry</a>. (When X=∅, the result is obvious. When X≠∅, the <a href="Reverse_triangle_inequality" class="mw-redirect" title="Reverse triangle inequality">reverse triangle inequality</a> implies the result.)</li>

<li>Let f in T(X). For any a in X, if f(a)=0, then f=e(a).<a href="#cite_note-DHKMS-5">[3]</a>: Lemma 5.1 (For every x in X we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e(a))(x)=d(a,x)\leq f(a)+f(x)=f(x).}">

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<annotation encoding="application/x-tex">{\displaystyle (e(a))(x)=d(a,x)\leq f(a)+f(x)=f(x).}</annotation>

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</math><img src="./47864a9e82c7a703df237da6190488801ee7cce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.625ex; height:2.843ex;" alt="{\displaystyle (e(a))(x)=d(a,x)\leq f(a)+f(x)=f(x).}" loading="lazy"> From minimality (second equivalent characterization in above section) of f and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(a)}">

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<annotation encoding="application/x-tex">{\displaystyle e(a)}</annotation>

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</math><img src="./692cb6d01ea86abbd4a3053561d1494b0967cd13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.123ex; height:2.843ex;" alt="{\displaystyle e(a)}" loading="lazy"> satisfies the first requirement it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=e_{a}.}">

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<annotation encoding="application/x-tex">{\displaystyle f=e_{a}.}</annotation>

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</math><img src="./0a97ab4e1ead5c36cbd8d722d3e8fb3b9b555770.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.209ex; height:2.509ex;" alt="{\displaystyle f=e_{a}.}" loading="lazy">)</li>

<li>(X,d) is <a href="Hyperbolic_metric_space" title="Hyperbolic metric space">hyperbolic</a> if and only if (T(X),δ) is hyperbolic.<a href="#cite_note-DHKMS-5">[3]</a>: Theorem 5.3 </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Hyperconvexity_properties">Hyperconvexity properties</h2></div>

<mrow>

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<mi>range</mi>

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<mi>range</mi>

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<annotation encoding="application/x-tex">{\displaystyle \left(X\cup (T(X)\setminus \operatorname {range} e),\delta _{(T(X)\setminus \operatorname {range} e)\times (T(X)\setminus \operatorname {range} e)}\cup (\delta (e(x),e(y)))_{x,y\in X}\cup (\delta (e(x),g))_{x\in X,g\in T(X)\setminus \operatorname {range} e}\cup (\delta (f,e(y))_{f\in T(X)\setminus \operatorname {range} e,y\in X}\right)}</annotation>

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</math> are both <a href="Injective_metric_space" title="Injective metric space">hyperconvex</a>.<a href="#cite_note-KK-2">[2]</a>: Proposition 4.7.1 </li>

<mi>range</mi>

<mi>range</mi>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {range} e\subseteq Y\subsetneq X\cup (T(X)\setminus \operatorname {range} e),}</annotation>

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</math><img src="./fc09a8a372ccb73193427047bf24903f4d4ead36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.673ex; height:2.843ex;" alt="{\displaystyle \operatorname {range} e\subseteq Y\subsetneq X\cup (T(X)\setminus \operatorname {range} e),}" loading="lazy"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X\cup (Y\setminus \operatorname {range} e),\delta _{(Y\setminus \operatorname {range} e)\times (Y\setminus \operatorname {range} e)}\cup (\delta (e(x),e(y)))_{x,y\in X}\cup (\delta (e(x),g))_{x\in X,g\in Y\setminus \operatorname {range} e}\cup (\delta (f,e(y))_{f\in Y\setminus \operatorname {range} e,y\in X}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(X\cup (Y\setminus \operatorname {range} e),\delta _{(Y\setminus \operatorname {range} e)\times (Y\setminus \operatorname {range} e)}\cup (\delta (e(x),e(y)))_{x,y\in X}\cup (\delta (e(x),g))_{x\in X,g\in Y\setminus \operatorname {range} e}\cup (\delta (f,e(y))_{f\in Y\setminus \operatorname {range} e,y\in X}\right)}</annotation>

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</math> is not hyperconvex.<a href="#cite_note-KK-2">[2]</a>: Proposition 4.7.2 ("(T(X),δ) is a hyperconvex hull of (X,d).")</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (H,\varepsilon )}</annotation>

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</math><img src="./4bcc1b64342389d009f2205adaea0166e4ae5270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.99ex; height:2.843ex;" alt="{\displaystyle (H,\varepsilon )}" loading="lazy"> be a hyperconvex metric space with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq H}">

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<annotation encoding="application/x-tex">{\displaystyle X\subseteq H}</annotation>

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</math><img src="./15daac8a881767d66dc543aa2d441e50a20f3fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.142ex; height:2.343ex;" alt="{\displaystyle X\subseteq H}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon |_{X\times X}=\delta }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon |_{X\times X}=\delta }</annotation>

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</math><img src="./8bcd07626e949345b51e58589cdba9a4197611f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.188ex; height:2.843ex;" alt="{\displaystyle \varepsilon |_{X\times X}=\delta }" loading="lazy">. If for all I with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq I\subsetneq H,}">

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<annotation encoding="application/x-tex">{\displaystyle X\subseteq I\subsetneq H,}</annotation>

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</math><img src="./19bd0adcddbdd31d691ceeb8df80dd103fb9ed8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.059ex; height:2.676ex;" alt="{\displaystyle X\subseteq I\subsetneq H,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (I,\varepsilon |_{I\times I})}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle (I,\varepsilon |_{I\times I})}</annotation>

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</math><img src="./dbce8fc17c80ee5a69f7358e43085e76ceb509e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.913ex; height:2.843ex;" alt="{\displaystyle (I,\varepsilon |_{I\times I})}" loading="lazy"> is not hyperconvex, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (H,\varepsilon )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (H,\varepsilon )}</annotation>

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</math><img src="./4bcc1b64342389d009f2205adaea0166e4ae5270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.99ex; height:2.843ex;" alt="{\displaystyle (H,\varepsilon )}" loading="lazy"> and (T(X),δ) are <a href="Isometry#Isometry_definition" title="Isometry">isometric</a>.<a href="#cite_note-KK-2">[2]</a>: Proposition 4.7.1 ("Every hyperconvex hull of (X,d) is isometric with (T(X),δ).")</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>Say |X|=3, choose distinct a, b, c such that X={a,b,c}, and let i=d(a,b), j=d(a,c), k=d(b,c). Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}T(X)=&\{v\in (\mathbb {R} _{\geq 0})^{3}:1=v_{a}+v_{b},2=v_{a}+v_{c},3\leq v_{b}+v_{c}\\&\qquad \qquad \qquad {\text{or }}1=v_{a}+v_{b},2\leq v_{a}+v_{c},3=v_{b}+v_{c}\\&\qquad \qquad \qquad {\text{or }}1\leq v_{a}+v_{b},2=v_{a}+v_{c},3=v_{b}+v_{c}\}\\=&\{v\in (\mathbb {R} _{\geq 0})^{3}:v_{a}\leq {\frac {(i+j)-k}{2}},v_{b}=i-v_{a},v_{c}=j-v_{a}\\&\qquad \qquad \qquad {\text{or }}v_{a}=i-v_{b},v_{b}\leq {\frac {(i+k)-j}{2}},v_{c}=k-v_{b}\\&\qquad \qquad \qquad {\text{or }}v_{a}=j-v_{c},v_{b}=k-v_{c},v_{c}\leq {\frac {(j+k)-i}{2}}\}\\=&\left\{(t,i-t,j-t):t\in \left[0,i\land j\land {\frac {(i+j)-k}{2}}\right]\right\}\\&\cup \left\{(i-t,t,k-t):t\in \left[0,i\land k\land {\frac {(i+k)-j}{2}}\right]\right\}\\&\cup \left\{(j-t,k-t,t):t\in \left[0,j\land k\land {\frac {(j+k)-i}{2}}\right]\right\}\\=&\left\{(t,i-t,j-t):t\in \left[0,{\frac {(i+j)-k}{2}}\right]\right\}\\&\cup \left\{(i-t,t,k-t):t\in \left[0,{\frac {(i+k)-j}{2}}\right]\right\}\\&\cup \left\{(j-t,k-t,t):t\in \left[0,{\frac {(j+k)-i}{2}}\right]\right\}\\=&\operatorname {conv} \{(0,i,j),x\}\cup \operatorname {conv} \{(i,0,k),x\}\cup \operatorname {conv} \{(j,k,0),x\},\end{alignedat}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}T(X)=&\{v\in (\mathbb {R} _{\geq 0})^{3}:1=v_{a}+v_{b},2=v_{a}+v_{c},3\leq v_{b}+v_{c}\\&\qquad \qquad \qquad {\text{or }}1=v_{a}+v_{b},2\leq v_{a}+v_{c},3=v_{b}+v_{c}\\&\qquad \qquad \qquad {\text{or }}1\leq v_{a}+v_{b},2=v_{a}+v_{c},3=v_{b}+v_{c}\}\\=&\{v\in (\mathbb {R} _{\geq 0})^{3}:v_{a}\leq {\frac {(i+j)-k}{2}},v_{b}=i-v_{a},v_{c}=j-v_{a}\\&\qquad \qquad \qquad {\text{or }}v_{a}=i-v_{b},v_{b}\leq {\frac {(i+k)-j}{2}},v_{c}=k-v_{b}\\&\qquad \qquad \qquad {\text{or }}v_{a}=j-v_{c},v_{b}=k-v_{c},v_{c}\leq {\frac {(j+k)-i}{2}}\}\\=&\left\{(t,i-t,j-t):t\in \left[0,i\land j\land {\frac {(i+j)-k}{2}}\right]\right\}\\&\cup \left\{(i-t,t,k-t):t\in \left[0,i\land k\land {\frac {(i+k)-j}{2}}\right]\right\}\\&\cup \left\{(j-t,k-t,t):t\in \left[0,j\land k\land {\frac {(j+k)-i}{2}}\right]\right\}\\=&\left\{(t,i-t,j-t):t\in \left[0,{\frac {(i+j)-k}{2}}\right]\right\}\\&\cup \left\{(i-t,t,k-t):t\in \left[0,{\frac {(i+k)-j}{2}}\right]\right\}\\&\cup \left\{(j-t,k-t,t):t\in \left[0,{\frac {(j+k)-i}{2}}\right]\right\}\\=&\operatorname {conv} \{(0,i,j),x\}\cup \operatorname {conv} \{(i,0,k),x\}\cup \operatorname {conv} \{(j,k,0),x\},\end{alignedat}}}</annotation>

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</math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2^{-1}(i+j-k,i+k-j,j+k-i).}">

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<annotation encoding="application/x-tex">{\displaystyle x=2^{-1}(i+j-k,i+k-j,j+k-i).}</annotation>

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</math><img src="./5ff1dd719c170a6bad299bee851512367d731a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.405ex; height:3.176ex;" alt="{\displaystyle x=2^{-1}(i+j-k,i+k-j,j+k-i).}" loading="lazy"> [Add a picture. Caption: If X={0,1,2}, then T(X)=conv{(,,),(,,)} u conv{(,,),(,,)} u conv{(,,),(,,)} is shaped like the letter Y.] (Cf. <a href="#cite_note-HRS-7">[4]</a>: 124 )</li></ul>

<ul><li>The figure shows a set X of 16 points in the plane; to form a finite metric space from these points, we use the <a href="Manhattan_distance" class="mw-redirect" title="Manhattan distance">Manhattan distance</a> (ℓ1 distance).<a href="#cite_note-13">[8]</a> The blue region shown in the figure is the <a href="Orthogonal_convex_hull" title="Orthogonal convex hull">orthogonal convex hull</a>, the set of points z such that each of the four closed quadrants with z as apex contains a point of X. Any such point z corresponds to a point of the tight span: the function f(x) corresponding to a point z is f(x) = d(z,x). A function of this form satisfies property 1 of the tight span for any z in the Manhattan-metric plane, by the triangle inequality for the Manhattan metric. To show property 2 of the tight span, consider some point x in X; we must find y in X such that f(x)+f(y)=d(x,y). But if x is in one of the four quadrants having z as apex, y can be taken as any point in the opposite quadrant, so property 2 is satisfied as well. Conversely it can be shown that every point of the tight span corresponds in this way to a point in the orthogonal convex hull of these points. However, for point sets with the Manhattan metric in higher dimensions, and for planar point sets with disconnected orthogonal hulls, the tight span differs from the orthogonal convex hull.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Dimension_of_the_tight_span_when_X_is_finite">Dimension of the tight span when X is finite</h2></div>

The definition above embeds the tight span T(X) of a set of n (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} _{\geq 0}}">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} _{\geq 0}}</annotation>

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</math><img src="./95800ce5aede1cb3e0bd5908ec2290c0dd491d7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.118ex; height:2.676ex;" alt="{\displaystyle n\in \mathbb {Z} _{\geq 0}}" loading="lazy">) points into RX, a real vector space of dimension n. On the other hand, if we consider the dimension of T(X) as a <a href="Polyhedral_complex" title="Polyhedral complex">polyhedral complex</a>, <a href="#CITEREFDevelin2006">Develin (2006)</a> showed that, with a suitable general position assumption on the metric, this definition leads to a space with dimension between n/3 and n/2.

<div class="mw-heading mw-heading2"><h2 id="Alternative_definitions">Alternative definitions</h2></div>

An alternative definition based on the notion of a <a href="Metric_space_aimed_at_its_subspace" title="Metric space aimed at its subspace">metric space aimed at its subspace</a> was described by <a href="#CITEREFHolsztyński1968">Holsztyński (1968)</a>, who proved that the injective envelope of a Banach space, in the category of Banach spaces, coincides (after forgetting the linear structure) with the tight span. This theorem allows to reduce certain problems from arbitrary Banach spaces to Banach spaces of the form C(X), where X is a compact space.

<a href="#CITEREFDevelinSturmfels2004">Develin & Sturmfels (2004)</a> attempted to provide an alternative definition of the tight span of a finite metric space as the <a href="Tropical_geometry" title="Tropical geometry">tropical convex hull</a> of the vectors of distances from each point to each other point in the space. However, later the same year they acknowledged in an Erratum <a href="#CITEREFDevelinSturmfels2004a">Develin & Sturmfels (2004a)</a> that, while the tropical convex hull always contains the tight span, it may not coincide with it.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<ul><li><a href="#CITEREFDressHuberMoulton2001">Dress, Huber & Moulton (2001)</a> describe applications of the tight span in <a href="Phylogenetics" title="Phylogenetics">reconstructing evolutionary trees</a> from biological data.</li>

<li>The tight span serves a role in several <a href="Online_algorithm" title="Online algorithm">online algorithms</a> for the <a href="K-server_problem" title="K-server problem">K-server problem</a>.<a href="#cite_note-14">[9]</a></li>

<li><a href="#CITEREFSturmfelsYu2004">Sturmfels & Yu (2004)</a> uses the tight span to classify metric spaces on up to six points.</li>

<li><a href="#CITEREFChepoi1997">Chepoi (1997)</a> uses the tight span to prove results about packing cut metrics into more general finite metric spaces.</li></ul>

<ul><li><a href="Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a>, an embedding of any metric space into a <a href="Banach_space" title="Banach space">Banach space</a> defined similarly to the Kuratowski map</li>

<li><a href="Injective_metric_space" title="Injective metric space">Injective metric space</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFDressHuberMoulton2001">Dress, Huber & Moulton (2001)</a>.

</li>

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</style><cite id="CITEREFKhamsiKirk2001" class="citation book cs1"><a href="Mohamed_Amine_Khamsi" title="Mohamed Amine Khamsi">Khamsi, Mohamed A.</a>; <a href="William_Arthur_Kirk" title="William Arthur Kirk">Kirk, William A.</a> (2001). An Introduction to Metric Spaces and Fixed Point Theory. Wiley.</cite>

</li>

<li id="cite_note-DHKMS-5">^ <a href="#cite_ref-DHKMS_5-0">a</a> <a href="#cite_ref-DHKMS_5-1">b</a> <a href="#cite_ref-DHKMS_5-2">c</a> <cite id="CITEREFDressHuberKoolenMoulton2012" class="citation book cs1"><a href="Andreas_Dress" title="Andreas Dress">Dress, Andreas</a>; <a href="Katharina_T._Huber" title="Katharina T. Huber">Huber, Katharina T.</a>; Koolen, Jacobus; Moulton, Vincent; Spillner, Andreas (2012). Basic Phylogenetic Combinatorics. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-76832-0</bdi>.</cite>

</li>

<li id="cite_note-HRS-7">^ <a href="#cite_ref-HRS_7-0">a</a> <a href="#cite_ref-HRS_7-1">b</a> <a href="#cite_ref-HRS_7-2">c</a> <cite id="CITEREFHusonRuppScornavacca2010" class="citation book cs1">Huson, Daniel H.; Rupp, Regula; Scornavacca, Celine (2010). Phylogenetic Networks: Conceps, Algorithms and Applications. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-75596-2</bdi>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFDezaDeza2014" class="citation book cs1"><a href="Michel_Deza" title="Michel Deza">Deza, Michel Marie</a>; <a href="Elena_Deza" title="Elena Deza">Deza, Elena</a> (2014). Encyclopedia of Distances (Third ed.). Springer. p. 47. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-44341-5</bdi>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFMelleray2008" class="citation journal cs1">Melleray, Julien (2008). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.topol.2007.04.029">"Some geometric and dynamical properties of the Urysohn space"</a>. Topology and Its Applications. 155 (14): 1531–1560. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.topol.2007.04.029">10.1016/j.topol.2007.04.029</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFBenyaminiLindenstrauss2000" class="citation book cs1"><a href="Yoav_Benjamini" title="Yoav Benjamini">Benyamini, Yoav</a>; <a href="Joram_Lindenstrauss" title="Joram Lindenstrauss">Lindenstrauss, Joram</a> (2000). Geometric Nonlinear Functional Analysis. American Mathematical Society. p. 32. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-0835-1</bdi>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> In two dimensions, the Manhattan distance is isometric after rotation and scaling to the <a href="Lp_space#General_ℓp-space" title="Lp space">ℓ∞ distance</a>, so with this metric the plane is itself injective, but this equivalence between ℓ1 and ℓ∞ does not hold in higher dimensions.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFChrobakLarmore1994">Chrobak & Larmore (1994)</a>.

</li>

</ol></div>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Khamsi and Kirk use this condition in their definition.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Khamsi and Kirk's proof shows one implication of the equivalence to the condition immediately above. The other implication is not difficult to show.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> I.e., the Kuratowski map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(x)\in T(X).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(x)\in T(X).}</annotation>

</semantics>

</math><img src="./eb0ce7fb9532b9506a1f5c69b74c66dcbb08863e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.136ex; height:2.843ex;" alt="{\displaystyle e(x)\in T(X).}" loading="lazy"> We will introduce the Kuratowski map below.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> The supremum is achieved with y=x.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> The supremum is achieved with y=x.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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<ul><li><cite id="CITEREFChepoi1997" class="citation cs2">Chepoi, Victor (1997), "A TX approach to some results on cuts and metrics", Advances in Applied Mathematics, 19 (4): 453–470, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Faama.1997.0549">10.1006/aama.1997.0549</a></cite>.</li>

<li><cite id="CITEREFChrobakLarmore1994" class="citation cs2"><a href="Marek_Chrobak" title="Marek Chrobak">Chrobak, Marek</a>; <a href="Lawrence_L._Larmore" title="Lawrence L. Larmore">Larmore, Lawrence L.</a> (1994), "Generosity helps or an 11-competitive algorithm for three servers", Journal of Algorithms, 16 (2): 234–263, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjagm.1994.1011">10.1006/jagm.1994.1011</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15169525">15169525</a></cite>.</li>

<li><cite id="CITEREFDevelin2006" class="citation cs2"><a href="Mike_Develin" title="Mike Develin">Develin, Mike</a> (2006), "Dimensions of tight spans", <a href="Annals_of_Combinatorics" title="Annals of Combinatorics">Annals of Combinatorics</a>, 10 (1): 53–61, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.CO/0407317">math.CO/0407317</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00026-006-0273-y">10.1007/s00026-006-0273-y</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:92984638">92984638</a></cite>.</li>

<li><cite id="CITEREFDevelinSturmfels2004" class="citation cs2"><a href="Mike_Develin" title="Mike Develin">Develin, Mike</a>; <a href="Bernd_Sturmfels" title="Bernd Sturmfels">Sturmfels, Bernd</a> (2004), <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/documenta/vol-09/01.pdf">"Tropical convexity"</a> (PDF), Documenta Mathematica, 9: 1–27, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2Fdm%2F154">10.4171/dm/154</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:64471">64471</a></cite>.</li>

<li><cite id="CITEREFDevelinSturmfels2004a" class="citation cs2"><a href="Mike_Develin" title="Mike Develin">Develin, Mike</a>; <a href="Bernd_Sturmfels" title="Bernd Sturmfels">Sturmfels, Bernd</a> (2004a), <a rel="nofollow" class="external text" href="https://www.math.uni-bielefeld.de/documenta/vol-09/12.pdf">"Erratum for "Tropical Convexity""</a> (PDF), Documenta Mathematica, 9: 205–206, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2Fdm%2F154">10.4171/dm/154</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:64471">64471</a></cite>.</li>

<li><cite id="CITEREFDress1984" class="citation cs2"><a href="Andreas_Dress" title="Andreas Dress">Dress, Andreas W. M.</a> (1984), "Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups", <a href="Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a>, 53 (3): 321–402, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2884%2990029-X">10.1016/0001-8708(84)90029-X</a></cite>.</li>

<li><cite id="CITEREFDressHuberMoulton2001" class="citation cs2"><a href="Andreas_Dress" title="Andreas Dress">Dress, Andreas W. M.</a>; <a href="Katharina_T._Huber" title="Katharina T. Huber">Huber, K. T.</a>; Moulton, V. (2001), <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/documenta/lsu/dress-huber-multon.pdf">"Metric spaces in pure and applied mathematics"</a> (PDF), Documenta Mathematica, Documenta Mathematica Series, 2 (Proceedings Quadratic Forms LSU): 121–139, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2Fdms%2F2%2F5">10.4171/dms/2/5</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-98547-042-6</bdi></cite>.</li>

<li><cite id="CITEREFHolsztyński1968" class="citation cs2">Holsztyński, Włodzimierz (1968), "Linearisation of isometric embeddings of Banach Spaces. Metric Envelopes.", Bull. Acad. Polon. Sci., 16: 189–193</cite>.</li>

<li><cite id="CITEREFIsbell1964" class="citation cs2"><a href="John_R._Isbell" title="John R. Isbell">Isbell, J. R.</a> (1964), "Six theorems about injective metric spaces", <a href="Comment._Math._Helv." class="mw-redirect" title="Comment. Math. Helv.">Comment. Math. Helv.</a>, 39: 65–76, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02566944">10.1007/BF02566944</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121857986">121857986</a></cite>.</li>

<li><cite id="CITEREFSturmfelsYu2004" class="citation cs2"><a href="Bernd_Sturmfels" title="Bernd Sturmfels">Sturmfels, Bernd</a>; Yu, Josephine (2004), <a rel="nofollow" class="external text" href="http://www.combinatorics.org/Volume_11/Abstracts/v11i1r44.html">"Classification of Six-Point Metrics"</a>, The Electronic Journal of Combinatorics, 11: R44, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.MG/0403147">math.MG/0403147</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004math......3147S">2004math......3147S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.37236%2F1797">10.37236/1797</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6733896">6733896</a></cite>.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFJoswig" class="citation cs2">Joswig, Michael, <a rel="nofollow" class="external text" href="http://homepages.math.tu-berlin.de/~joswig/tightspans/index.html">Tight spans</a></cite>.</li></ul>

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</style><div id="Metric_spaces_(Category)86" style="font-size:114%;margin:0 4em"><a href="Metric_space" title="Metric space">Metric spaces</a> (Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Metric_space" title="Metric space">Metric space</a></li>

<li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Complete_metric_space" title="Complete metric space">Completeness</a></li>

<li><a href="Equivalence_of_metrics" title="Equivalence of metrics">Equivalent metrics</a></li>

<li><a href="Metrizable_space" title="Metrizable space">Metrizable space</a></li>

<li><a href="Triangle_inequality" title="Triangle inequality">Triangle inequality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Baire_category_theorem" title="Baire category theorem">Baire category theorem</a></li>

<li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point</a></li>

<li><a href="Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a></li>

<li><a href="Lebesgue's_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a></li>

<li><a href="Metrization_theorem" class="mw-redirect" title="Metrization theorem">Metrization theorems</a>:

<li><a href="Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov</a></li>

<li><a href="Urysohn's_metrization_theorem" class="mw-redirect" title="Urysohn's metrization theorem">Urysohn's</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contraction_mapping" title="Contraction mapping">Contraction</a>

<ul><li><a href="Metric_map" title="Metric map">Metric map</a></li></ul></li>

<li><a href="Dilation_(metric_space)" title="Dilation (metric space)">Dilation</a></li>

<li><a href="Equicontinuity" title="Equicontinuity">Equicontinuity</a></li>

<li>(<a href="Quasi-isometry" title="Quasi-isometry">Quasi-</a>) <a href="Isometry" title="Isometry">Isometry</a></li>

<li><a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a></li>

<li><a href="Metric_derivative" title="Metric derivative">Metric derivative</a></li>

<li><a href="Metric_outer_measure" title="Metric outer measure">Metric outer measure</a></li>

<li><a href="Metric_projection" title="Metric projection">Metric projection</a></li>

<li><a href="Motion_(geometry)" title="Motion (geometry)">Motion</a></li>

<li><a href="Quasisymmetric_map" title="Quasisymmetric map">Quasisymmetric</a></li>

<li><a href="Stretch_factor" title="Stretch factor">Stretch factor</a></li>

<li><a href="Uniform_continuity" title="Uniform continuity">Uniform continuity</a>

<ul><li><a href="Uniform_isomorphism" title="Uniform isomorphism">Isomorphism</a></li></ul></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform convergence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of metric spaces</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Complete_metric_space" title="Complete metric space">Complete</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex</a></li>

<li><a href="Doubling_space" title="Doubling space">Doubling</a></li>

<li><a href="Hyperbolic_metric_space" title="Hyperbolic metric space">Hyperbolic</a></li>

<li><a href="Injective_metric_space" title="Injective metric space">Injective</a></li>

<li><a href="Length_metric_space" class="mw-redirect" title="Length metric space">Length metric space</a></li>

<li><a href="Metric_space_aimed_at_its_subspace" title="Metric space aimed at its subspace">Metric space aimed at its subspace</a></li>

<li><a href="Polish_space" title="Polish space">Polish</a></li>

<li><a href="Totally_bounded_space" title="Totally bounded space">Totally bounded</a></li>

<li><a href="Tree-graded_space" title="Tree-graded space">Tree-graded</a></li>

<li><a href="Ultrametric_space" title="Ultrametric space">Ultrametric space</a></li>

<li><a href="Uniformly_disconnected_space" title="Uniformly disconnected space">Uniformly disconnected</a></li>

<li><a href="Urysohn_universal_space" title="Urysohn universal space">Urysohn universal</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ball_(mathematics)" title="Ball (mathematics)">Balls</a></li>

<li><a href="Borel_set" title="Borel set">Borel</a></li>

<li><a href="Bounded_set" title="Bounded set">Bounded</a></li>

<li><a href="Delone_set" title="Delone set">Delone</a></li>

<li><a href="Diameter_of_a_set" title="Diameter of a set">Diameter</a></li>

<li><a href="Distance_set" title="Distance set">Distance set</a></li>

<li><a href="Gromov_product" title="Gromov product">Gromov product</a></li>

<li><a href="Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff convergence</a></li>

<li><a href="Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a></li>

<li><a href="Kuratowski_convergence" title="Kuratowski convergence">Kuratowski convergence</a></li>

<li><a href="Meyer_set" title="Meyer set">Meyer</a></li>

<li><a href="Packing_dimension" title="Packing dimension">Packing dimension</a></li>

<li><a href="Porous_set" title="Porous set">Porous</a></li>

<li><a href="Positively_separated_sets" title="Positively separated sets">Positively separated sets</a></li>

</ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Manifold" title="Manifold">Manifolds</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Functional_analysis" title="Functional analysis">Functional analysis</a> and <a href="Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li>

<li><a href="Inner_product_space" title="Inner product space">Inner product space</a></li>

<li><a href="L%C3%A9vy_metric" title="Lévy metric">Lévy metric</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable topological vector space</a></li>

<li><a href="Normed_space" class="mw-redirect" title="Normed space">Normed space</a></li>

<li><a href="Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li>

<li><a href="Wasserstein_metric" title="Wasserstein metric">Wasserstein metric</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="General_topology" title="General topology">General topology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Discrete_space" title="Discrete space">Discrete space</a></li>

<li><a href="Intrinsic_metric" title="Intrinsic metric">Intrinsic metric</a></li>

<li><a href="Laakso_space" title="Laakso space">Laakso space</a></li>

<li><a href="Product_metric" title="Product metric">Product metric</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Category_of_metric_spaces" title="Category of metric spaces">Category of metric spaces</a></li>

<li><a href="Cantor_space" title="Cantor space">Cantor space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approach_space" title="Approach space">Approach space</a></li>

<li><a href="Cauchy_space" title="Cauchy space">Cauchy space</a></li>

<li><a href="Coarse_structure" title="Coarse structure">Coarse structure</a></li>

<li><a href="Cosmic_space" title="Cosmic space">Cosmic space</a></li>

<li><a href="Diversity_(mathematics)" title="Diversity (mathematics)">Diversity</a></li>

<li><a href="Generalised_metric" title="Generalised metric">Generalised metric</a></li>

<li><a href="Measure_space" title="Measure space">Measure space</a></li>

<li><a href="Probabilistic_metric_space" title="Probabilistic metric space">Probabilistic metric space</a></li>

<li><a href="Proximity_space" title="Proximity space">Proximity space</a></li>

<li><a href="Pseudometric_space" title="Pseudometric space">Pseudometric space</a></li>

<li><a href="Uniform_space" title="Uniform space">Uniform space</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-08" href="https://en.wikipedia.org/wiki/?title=Tight_span&oldid=1284651140">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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