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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Orthant</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>Orthant</b> bezeichnet in der <a href="Geometrie" title="Geometrie">Geometrie</a> die <a href="Mengenlehre#Teilmenge" title="Mengenlehre">Teilmenge</a> des <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span>-<a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">dimensionalen</a> Raumes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a713426956296f1668fce772df3c60b9dde8a685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}" loading="lazy"></span>, die auf jeweils genau einer Seite der durch den <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Ursprung</a> verlaufenden <a href="Parallel_(Geometrie)" class="mw-redirect" title="Parallel (Geometrie)">achsenparallelen</a> <a href="Hyperebene" title="Hyperebene">Hyperebenen</a> liegt.
Ein Orthant ist damit eine Menge der Form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x=(x_{1},\ldots ,x_{d})\in \mathbb {R} ^{d}\mid x_{i}\cdot d_{i}\geq 0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
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<mi>x</mi>
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<mn>1</mn>
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<mi>x</mi>
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<mi>d</mi>
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<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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<mo>∣<!-- ∣ --></mo>
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<mi>x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
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<annotation encoding="application/x-tex">{\displaystyle \{x=(x_{1},\ldots ,x_{d})\in \mathbb {R} ^{d}\mid x_{i}\cdot d_{i}\geq 0\}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98b0b94f62bc41cfbd0a7e3e50d42ce879d15c5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.172ex; height:3.176ex;" alt="{\displaystyle \{x=(x_{1},\ldots ,x_{d})\in \mathbb {R} ^{d}\mid x_{i}\cdot d_{i}\geq 0\}}" loading="lazy"></span>,</dd></dl>
<p>wobei die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}\in \{-1,1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d_{i}\in \{-1,1\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce643f936b43dca5278c8cd4d1de5cfb5ac93c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.341ex; height:2.843ex;" alt="{\displaystyle d_{i}\in \{-1,1\}}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,\ldots ,d\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>d</mi>
<mo fence="false" stretchy="false">}</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,d\}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05658c5d2c00ef453c2df280b81d734908a1690e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.525ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,d\}}" loading="lazy"></span> fest gewählt sind.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Daraus folgt, dass es genau <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{d}}">
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<mrow class="MJX-TeXAtom-ORD">
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<mn>2</mn>
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<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle 2^{d}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e78df590f3fc81f0201082eaaa6844c145c8bdf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.254ex; height:2.676ex;" alt="{\displaystyle 2^{d}}" loading="lazy"></span> Orthanten gibt.
Genauer spricht man von abgeschlossenen Orthanten, denn es handelt sich um abgeschlossene Mengen. Die offenen Orthanten erhält man, wenn man in obiger Definition das <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq }">
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<mrow class="MJX-TeXAtom-ORD">
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<mo>≥<!-- ≥ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \geq }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \geq }" loading="lazy"></span> durch die strikte Ungleichung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>></mo>
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<annotation encoding="application/x-tex">{\displaystyle >}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b27b77ab4e3293ea9ce65cef60fea655c398423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}" loading="lazy"></span> ersetzt.
</p><p>Manche Autoren betrachten auch um einen festen Vektor verschobene Orthanten. So wird in<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in \mathbb {R} ^{d}\mid x\leq a\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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</msup>
<mo>∣<!-- ∣ --></mo>
<mi>x</mi>
<mo>≤<!-- ≤ --></mo>
<mi>a</mi>
<mo fence="false" stretchy="false">}</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} ^{d}\mid x\leq a\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74fd277f31fd8cd1d304ed298a93398d62c751b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.86ex; height:3.176ex;" alt="{\displaystyle \{x\in \mathbb {R} ^{d}\mid x\leq a\}}" loading="lazy"></span> als der untere Orthant an den Vektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} ^{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} ^{d}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fa432ab537420464914f4261a0462de154494d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.841ex; height:2.676ex;" alt="{\displaystyle a\in \mathbb {R} ^{d}}" loading="lazy"></span> bezeichnet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<ul><li>Im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span> sind die Orthanten die vier <a href="Quadrant_(Mathematik)" title="Quadrant (Mathematik)">Quadranten</a> des <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatensystems</a>.</li></ul>
<ul><li>Im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"></span> nennt man die acht Orthanten entsprechend <a href="Oktant_(Geometrie)" title="Oktant (Geometrie)">Oktanten</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Oliver Aberth: <i>Introduction to Precise Numerical Methods</i>, Elsevier 2007, ISBN 0-12-373859-8, Seite 142</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Branko Grünbaum: <i>Convex Polytopes</i>, 2-te Auflage, Springer 2003, Graduate Texts in Mathematics, ISBN 978-0-387-40409-7, Seite 305</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Rainer Dycherhoff, Karl Mosler: <i>Orthant orderings of discrete random vectors</i>, Journal of Statistical Planning and Inference 62 (1997), Seiten 193–205</span>
</li>
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