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Support function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Support_curve" class="mw-redirect" title="Support curve">Support curve</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the support function hA of a non-empty <a href="Closed_set" title="Closed set">closed</a> <a href="Convex_set" title="Convex set">convex set</a> A in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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describes the (signed) distances of <a href="Supporting_hyperplane" title="Supporting hyperplane">supporting hyperplanes</a> of A from the origin. The support function is a <a href="Convex_function" title="Convex function">convex function</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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Any non-empty closed convex set A is uniquely determined by hA. Furthermore, the support function, as a function of the set A, is compatible with many natural geometric operations, like scaling, translation, rotation and <a href="Minkowski_addition" title="Minkowski addition">Minkowski addition</a>.

Due to these properties, the support function is one of the most central basic concepts in convex geometry.

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

The support function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{A}\colon \mathbb {R} ^{n}\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle h_{A}\colon \mathbb {R} ^{n}\to \mathbb {R} }</annotation>

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</math><img src="./e30797050b128f9b3a339fc1be37bbbf8d775383.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.026ex; height:2.676ex;" alt="{\displaystyle h_{A}\colon \mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy">

of a non-empty closed convex set A in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> is given by

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<annotation encoding="application/x-tex">{\displaystyle h_{A}(x)=\sup\{x\cdot a:a\in A\},}</annotation>

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</math><img src="./2a8915c2945590bdb1f71d45832f45d724a23242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.504ex; height:2.843ex;" alt="{\displaystyle h_{A}(x)=\sup\{x\cdot a:a\in A\},}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation>

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.<a href="#cite_note-schneider-3">[3]</a> Its interpretation is most intuitive when x is a unit vector:

by definition, A is contained in the closed half space

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<annotation encoding="application/x-tex">{\displaystyle \{y\in \mathbb {R} ^{n}:y\cdot x\leqslant h_{A}(x)\}}</annotation>

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</math><img src="./8ec1cb61282b7b52082fd99ac3f55e04da23c3d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.36ex; height:2.843ex;" alt="{\displaystyle \{y\in \mathbb {R} ^{n}:y\cdot x\leqslant h_{A}(x)\}}" loading="lazy"></dd></dl>

and there is at least one point of A in the boundary

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<annotation encoding="application/x-tex">{\displaystyle H(x)=\{y\in \mathbb {R} ^{n}:y\cdot x=h_{A}(x)\}}</annotation>

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</math><img src="./cdfe88fe7052f5692259bbef6d991620c52af48f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.661ex; height:2.843ex;" alt="{\displaystyle H(x)=\{y\in \mathbb {R} ^{n}:y\cdot x=h_{A}(x)\}}" loading="lazy"></dd></dl>

of this half space. The hyperplane H(x) is therefore called a supporting hyperplane

with exterior (or outer) unit normal vector x.

The word exterior is important here, as

the orientation of x plays a role, the set H(x) is in general different from H(−x).

Now hA(x) is the (signed) distance of H(x) from the origin.

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

The support function of a singleton A = {a} is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{A}(x)=x\cdot a}">

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

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</math><img src="./eb8a30ca554c5c02b1884b489ab43e37ea75fe62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.28ex; height:2.843ex;" alt="{\displaystyle h_{A}(x)=x\cdot a}" loading="lazy">.

The support function of the Euclidean unit ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{y\in \mathbb {R} ^{n}\,:\,\|y\|_{2}\leq 1\}}">

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<annotation encoding="application/x-tex">{\displaystyle B=\{y\in \mathbb {R} ^{n}\,:\,\|y\|_{2}\leq 1\}}</annotation>

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</math><img src="./ac9a728d809bf31cf09a079a92bb1be39b69679d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.587ex; height:2.843ex;" alt="{\displaystyle B=\{y\in \mathbb {R} ^{n}\,:\,\|y\|_{2}\leq 1\}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{B}(x)=\|x\|_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{B}(x)=\|x\|_{2}}</annotation>

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</math><img src="./9054dd8943e6ad277bd1ad3402cd6612952ef158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.765ex; height:2.843ex;" alt="{\displaystyle h_{B}(x)=\|x\|_{2}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{2}}</annotation>

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</math><img src="./b3a8e44a2eb980f856968a6357e3d0a7c22c905f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{2}}" loading="lazy"> is the 2-norm.

If A is a line segment through the origin with endpoints −a and a, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{A}(x)=|x\cdot a|}">

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<annotation encoding="application/x-tex">{\displaystyle h_{A}(x)=|x\cdot a|}</annotation>

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</math><img src="./ff4ac9e3918ed51b7a2795bc114485880733caff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.574ex; height:2.843ex;" alt="{\displaystyle h_{A}(x)=|x\cdot a|}" loading="lazy">.

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

<div class="mw-heading mw-heading3"><h3 id="As_a_function_of_x">As a function of x</h3></div>

The support function of a compact nonempty convex set is real valued and continuous, but if the

set is closed and unbounded, its support function is extended real valued (it takes the value

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

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</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">). As any nonempty closed convex set is the intersection of

its supporting half spaces, the function hA determines A uniquely.

This can be used to describe certain geometric properties of convex sets analytically.

For instance, a set A is point symmetric with respect to the origin if and only if hA

is an <a href="Even_function" class="mw-redirect" title="Even function">even function</a>.

In general, the support function is not differentiable.

However, directional derivatives exist and yield support functions of support sets. If A is compact and convex,

and hA'(u;x) denotes the directional derivative of

hA at u ≠ 0 in direction x,

we have

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<annotation encoding="application/x-tex">{\displaystyle h_{A}'(u;x)=h_{A\cap H(u)}(x)\qquad x\in \mathbb {R} ^{n}.}</annotation>

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</math><img src="./602f5f4a3363b32f40293a77abc3e77bb4b4174f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.482ex; height:3.176ex;" alt="{\displaystyle h_{A}'(u;x)=h_{A\cap H(u)}(x)\qquad x\in \mathbb {R} ^{n}.}" loading="lazy"></dd></dl>

Here H(u) is the supporting hyperplane of A with exterior normal vector u, defined

above. If A ∩ H(u) is a singleton {y}, say, it follows that the support function is differentiable at

u and its gradient coincides with y. Conversely, if hA is differentiable at u, then A ∩ H(u) is a singleton. Hence hA is differentiable at all points u ≠ 0

if and only if A is strictly convex (the boundary of A does not contain any line segments).

More generally, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is convex and closed then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \mathbb {R} ^{n}\setminus \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle u\in \mathbb {R} ^{n}\setminus \{0\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \partial h_{A}(u)=H(u)\cap A\,,}</annotation>

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</math><img src="./4f80f7322369812a199ae7adf469810fdf6bec95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.922ex; height:2.843ex;" alt="{\displaystyle \partial h_{A}(u)=H(u)\cap A\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial h_{A}(u)}">

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<annotation encoding="application/x-tex">{\displaystyle \partial h_{A}(u)}</annotation>

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</math><img src="./3f613ab7161e6aab1cddee287fe0387ae91c7ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.261ex; height:2.843ex;" alt="{\displaystyle \partial h_{A}(u)}" loading="lazy"> denotes the set of <a href="Subgradient" class="mw-redirect" title="Subgradient">subgradients</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle h_{A}}</annotation>

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</math><img src="./70d9fb28189a429e3e9675911aa921326a217b1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle h_{A}}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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It follows directly from its definition that the support function is positive homogeneous:

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<annotation encoding="application/x-tex">{\displaystyle h_{A}(\alpha x)=\alpha h_{A}(x),\qquad \alpha \geq 0,x\in \mathbb {R} ^{n},}</annotation>

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</math><img src="./1fe23631b8500e6f66f84b6d54ddfe8e29fc5663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.135ex; height:2.843ex;" alt="{\displaystyle h_{A}(\alpha x)=\alpha h_{A}(x),\qquad \alpha \geq 0,x\in \mathbb {R} ^{n},}" loading="lazy"></dd></dl>

and subadditive:

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<annotation encoding="application/x-tex">{\displaystyle h_{A}(x+y)\leq h_{A}(x)+h_{A}(y),\qquad x,y\in \mathbb {R} ^{n}.}</annotation>

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</math><img src="./64ca88d2c79bccd5b5f5957a8126bde39a1315be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.171ex; height:2.843ex;" alt="{\displaystyle h_{A}(x+y)\leq h_{A}(x)+h_{A}(y),\qquad x,y\in \mathbb {R} ^{n}.}" loading="lazy"></dd></dl>

It follows that hA is a <a href="Convex_function" title="Convex function">convex function</a>.

It is crucial in convex geometry that these properties characterize support functions:

Any positive homogeneous, convex, real valued function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> is the

support function of a nonempty compact convex set. Several proofs are known, <a href="#cite_note-schneider-3">[3]</a>

one is using the fact that the <a href="Legendre_transformation" title="Legendre transformation">Legendre transform</a> of a positive homogeneous, convex, real valued function

is the (convex) indicator function of a compact convex set.

Many authors restrict the support function to the Euclidean unit sphere

and consider it as a function on Sn-1.

The homogeneity property shows that this restriction determines the

support function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="As_a_function_of_A">As a function of A</h3></div>

The support functions of a dilated or translated set are closely related to the original set A:

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<annotation encoding="application/x-tex">{\displaystyle h_{\alpha A}(x)=\alpha h_{A}(x),\qquad \alpha \geq 0,x\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./31b8071199ec83f1d36ec667292caf2e1ab7d8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.052ex; height:2.843ex;" alt="{\displaystyle h_{\alpha A}(x)=\alpha h_{A}(x),\qquad \alpha \geq 0,x\in \mathbb {R} ^{n}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle h_{A+b}(x)=h_{A}(x)+x\cdot b,\qquad x,b\in \mathbb {R} ^{n}.}</annotation>

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</math><img src="./fee8e7dc9211749757c7a63ce9cb3705d6d70e74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.239ex; height:2.843ex;" alt="{\displaystyle h_{A+b}(x)=h_{A}(x)+x\cdot b,\qquad x,b\in \mathbb {R} ^{n}.}" loading="lazy"></dd></dl>

The latter generalises to

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<annotation encoding="application/x-tex">{\displaystyle h_{A+B}(x)=h_{A}(x)+h_{B}(x),\qquad x\in \mathbb {R} ^{n},}</annotation>

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</math><img src="./385ad105926c8ba3de650e48ba7845de977e27ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.701ex; height:2.843ex;" alt="{\displaystyle h_{A+B}(x)=h_{A}(x)+h_{B}(x),\qquad x\in \mathbb {R} ^{n},}" loading="lazy"></dd></dl>

where A + B denotes the <a href="Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sum</a>:

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<annotation encoding="application/x-tex">{\displaystyle A+B:=\{\,a+b\in \mathbb {R} ^{n}\mid a\in A,\ b\in B\,\}.}</annotation>

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</math><img src="./84f32e007152eebe46c5e156fc1c683aa69eab22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.611ex; height:2.843ex;" alt="{\displaystyle A+B:=\{\,a+b\in \mathbb {R} ^{n}\mid a\in A,\ b\in B\,\}.}" loading="lazy"></dd></dl>

The <a href="Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a> d H(A, B)

of two nonempty compact convex sets A and B can be expressed in terms of support functions,

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<annotation encoding="application/x-tex">{\displaystyle d_{\mathrm {H} }(A,B)=\|h_{A}-h_{B}\|_{\infty }}</annotation>

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</math><img src="./ddb6a8c400333c13f4dca465067054a79bb747dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.786ex; height:2.843ex;" alt="{\displaystyle d_{\mathrm {H} }(A,B)=\|h_{A}-h_{B}\|_{\infty }}" loading="lazy"></dd></dl>

where, on the right hand side, the <a href="Uniform_norm" title="Uniform norm">uniform norm</a> on the unit sphere is used.

The properties of the support function as a function of the set A are sometimes summarized in saying

that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">:A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mapsto }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bc09de045e7d82eef9fe078e7e7606576640c11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \mapsto }" loading="lazy"> h A maps the family of non-empty

compact convex sets to the cone of all real-valued continuous functions on the sphere whose positive

homogeneous extension is convex. Abusing terminology slightly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">

is sometimes called linear, as it respects Minkowski addition, although it is not

defined on a linear space, but rather on an (abstract) convex cone of nonempty compact convex sets.

The mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> is an isometry between this cone, endowed with the Hausdorff metric, and

a subcone of the family of continuous functions on Sn-1 with the uniform norm.

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

In contrast to the above, support functions are sometimes defined on the boundary of A rather than on

Sn-1, under the assumption that there exists a unique exterior unit normal at each boundary point.

Convexity is not needed for the definition.

For an oriented <a href="Smooth_surface" class="mw-redirect" title="Smooth surface">regular surface</a>, M, with a <a href="Unit_normal_vector" class="mw-redirect" title="Unit normal vector">unit normal vector</a>, N, defined everywhere on its surface, the support function

is then defined by

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {x}\mapsto {x}\cdot N({x})}</annotation>

</semantics>

</math><img src="./c60f5d086d3238e6854eefd64cb462d9146c16e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.155ex; height:2.843ex;" alt="{\displaystyle {x}\mapsto {x}\cdot N({x})}" loading="lazy">.</dd></dl>

In other words, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x}\in M}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {x}\in M}</annotation>

</semantics>

</math><img src="./07bcd86a2f3ae100d03e530aba4a38d28cb37a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle {x}\in M}" loading="lazy">, this support function gives the

signed distance of the unique hyperplane that touches M in x.

<ul><li><a href="Barrier_cone" title="Barrier cone">Barrier cone</a></li>

<li><a href="Supporting_functional" title="Supporting functional">Supporting functional</a></li></ul>

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

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<li id="cite_note-bonnesen-1"><a href="#cite_ref-bonnesen_1-0">^</a> T. Bonnesen, W. Fenchel, Theorie der konvexen Körper, Julius Springer, Berlin, 1934.

English translation: Theory of convex bodies, BCS Associates, Moscow, ID, 1987.

</li>

<li id="cite_note-gardner-2"><a href="#cite_ref-gardner_2-0">^</a> R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995. Second edition: 2006.

</li>

<li id="cite_note-schneider-3">^ <a href="#cite_ref-schneider_3-0">a</a> <a href="#cite_ref-schneider_3-1">b</a> R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.

</li>

</ol></div></div></div><div><div class="zim-footer">

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