<?xml version="1.0" encoding="UTF-8"?>
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<title><text class="ltx_LaTeX_logo" cssstyle="letter-spacing:-0.2em; margin-right:0.1em">L<text cssstyle="font-variant:small-caps;" yoffset="0.4ex">a</text>T<text cssstyle="font-variant:small-caps;font-size:120%" yoffset="-0.2ex">e</text>X</text>’s Newtheorem</title>
<date role="creation">none</date>
<section inlist="toc" xml:id="S1">
<tags>
<tag>1</tag>
<tag role="refnum">1</tag>
<tag role="typerefnum">§1</tag>
</tags>
<title><tag close=" ">1</tag>Test of standard theorem styles</title>
<theorem class="ltx_theorem_lem" inlist="thm theorem:lem" xml:id="S1.Thmthm1">
<tags>
<tag>Lemma 1.1</tag>
<tag role="refnum">1.1</tag>
<tag role="typerefnum">Lemma 1.1</tag>
</tags>
<title class="ltx_runin"><tag><text font="bold">Lemma 1.1</text></tag><text font="bold"> (negatively curved families)</text></title>
<para xml:id="S1.Thmthm1.p1">
<p><text font="italic">Let <Math mode="inline" tex="\{ds_{1}^{2},\dots,ds_{k}^{2}\}" text="set@(d * (s _ 1) ^ 2, dots, d * (s _ k) ^ 2)" xml:id="S1.Thmthm1.p1.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="set"/>
<XMRef idref="S1.Thmthm1.p1.m1.2"/>
<XMRef idref="S1.Thmthm1.p1.m1.1"/>
<XMRef idref="S1.Thmthm1.p1.m1.3"/>
</XMApp>
<XMWrap>
<XMTok font="upright" role="OPEN" stretchy="false">{</XMTok>
<XMApp xml:id="S1.Thmthm1.p1.m1.2">
<XMTok font="upright" meaning="times" role="MULOP"></XMTok>
<XMTok role="UNKNOWN">d</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok role="UNKNOWN">s</XMTok>
<XMTok font="upright" fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
</XMApp>
<XMTok font="upright" fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMTok font="upright" role="PUNCT">,</XMTok>
<XMTok font="upright" name="dots" role="ID" xml:id="S1.Thmthm1.p1.m1.1">…</XMTok>
<XMTok font="upright" role="PUNCT">,</XMTok>
<XMApp xml:id="S1.Thmthm1.p1.m1.3">
<XMTok font="upright" meaning="times" role="MULOP"></XMTok>
<XMTok role="UNKNOWN">d</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok role="UNKNOWN">s</XMTok>
<XMTok fontsize="70%" role="UNKNOWN">k</XMTok>
</XMApp>
<XMTok font="upright" fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMTok font="upright" role="CLOSE" stretchy="false">}</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math> be a negatively curved family of metrics
on <Math mode="inline" tex="\mathbf{D}_{r}" text="D _ r" xml:id="S1.Thmthm1.p1.m2">
<XMath>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="bold upright" role="UNKNOWN">D</XMTok>
<XMTok fontsize="70%" role="UNKNOWN">r</XMTok>
</XMApp>
</XMath>
</Math>, with associated forms <Math mode="inline" tex="\omega^{1}" text="omega ^ 1" xml:id="S1.Thmthm1.p1.m3">
<XMath>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok name="omega" role="UNKNOWN">ω</XMTok>
<XMTok font="upright" fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
</XMApp>
</XMath>
</Math>, …, <Math mode="inline" tex="\omega^{k}" text="omega ^ k" xml:id="S1.Thmthm1.p1.m4">
<XMath>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok name="omega" role="UNKNOWN">ω</XMTok>
<XMTok fontsize="70%" role="UNKNOWN">k</XMTok>
</XMApp>
</XMath>
</Math>.
Then <Math mode="inline" tex="\omega^{i}\leq\omega_{r}" text="omega ^ i <= omega _ r" xml:id="S1.Thmthm1.p1.m5">
<XMath>
<XMApp>
<XMTok font="upright" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok name="omega" role="UNKNOWN">ω</XMTok>
<XMTok fontsize="70%" role="UNKNOWN">i</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok name="omega" role="UNKNOWN">ω</XMTok>
<XMTok fontsize="70%" role="UNKNOWN">r</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math> for all <Math mode="inline" tex="i" text="i" xml:id="S1.Thmthm1.p1.m6">
<XMath>
<XMTok role="UNKNOWN">i</XMTok>
</XMath>
</Math>.</text></p>
</para>
</theorem>
<para xml:id="S1.p1">
<p>Then our main theorem:</p>
</para>
<theorem class="ltx_theorem_thm" inlist="thm theorem:thm" labels="LABEL:pigspan" xml:id="S1.Thmthm2">
<tags>
<tag>Theorem 1.2</tag>
<tag role="refnum">1.2</tag>
<tag role="typerefnum">Theorem 1.2</tag>
</tags>
<title class="ltx_runin"><tag><text font="bold">Theorem 1.2</text></tag></title>
<para xml:id="S1.Thmthm2.p1">
<p><text font="italic">Let <Math mode="inline" tex="d_{\max}" text="d _ maximum" xml:id="S1.Thmthm2.p1.m1">
<XMath>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok role="UNKNOWN">d</XMTok>
<XMTok font="upright" fontsize="70%" meaning="maximum" role="OPFUNCTION" scriptpos="post">max</XMTok>
</XMApp>
</XMath>
</Math> and <Math mode="inline" tex="d_{\min}" text="d _ minimum" xml:id="S1.Thmthm2.p1.m2">
<XMath>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok role="UNKNOWN">d</XMTok>
<XMTok font="upright" fontsize="70%" meaning="minimum" role="OPFUNCTION" scriptpos="post">min</XMTok>
</XMApp>
</XMath>
</Math> be the maximum, resp. minimum distance
between any two adjacent vertices of a quadrilateral <Math mode="inline" tex="Q" text="Q" xml:id="S1.Thmthm2.p1.m3">
<XMath>
<XMTok role="UNKNOWN">Q</XMTok>
</XMath>
</Math>. Let <Math mode="inline" tex="\sigma" text="sigma" xml:id="S1.Thmthm2.p1.m4">
<XMath>
<XMTok name="sigma" role="UNKNOWN">σ</XMTok>
</XMath>
</Math>
be the diagonal pigspan of a pig <Math mode="inline" tex="P" text="P" xml:id="S1.Thmthm2.p1.m5">
<XMath>
<XMTok role="UNKNOWN">P</XMTok>
</XMath>
</Math> with four legs.
Then <Math mode="inline" tex="P" text="P" xml:id="S1.Thmthm2.p1.m6">
<XMath>
<XMTok role="UNKNOWN">P</XMTok>
</XMath>
</Math> is capable of standing on the corners of <Math mode="inline" tex="Q" text="Q" xml:id="S1.Thmthm2.p1.m7">
<XMath>
<XMTok role="UNKNOWN">Q</XMTok>
</XMath>
</Math> iff</text></p>
<equation labels="LABEL:sdq" xml:id="S1.E1">
<tags>
<tag>(1)</tag>
<tag role="refnum">1</tag>
</tags>
<Math mode="display" tex="\sigma\geq\sqrt{d_{\max}^{2}+d_{\min}^{2}}." text="sigma >= square-root@((d _ maximum) ^ 2 + (d _ minimum) ^ 2)" xml:id="S1.E1.m1">
<XMath>
<XMDual>
<XMRef idref="S1.E1.m1.1"/>
<XMWrap>
<XMApp xml:id="S1.E1.m1.1">
<XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" name="sigma" role="UNKNOWN">σ</XMTok>
<XMApp>
<XMTok meaning="square-root"/>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok fontsize="70%" meaning="maximum" role="OPFUNCTION" scriptpos="post">max</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok fontsize="70%" meaning="minimum" role="OPFUNCTION" scriptpos="post">min</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMTok role="PERIOD">.</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
</theorem>
<theorem class="ltx_theorem_cor" inlist="thm theorem:cor" xml:id="S1.Thmthm3">
<tags>
<tag>Corollary 1.3</tag>
<tag role="refnum">1.3</tag>
<tag role="typerefnum">Corollary 1.3</tag>
</tags>
<title class="ltx_runin"><tag><text font="bold">Corollary 1.3</text></tag></title>
<para xml:id="S1.Thmthm3.p1">
<p><text font="italic">Admitting reflection and rotation, a three-legged pig <Math mode="inline" tex="P" text="P" xml:id="S1.Thmthm3.p1.m1">
<XMath>
<XMTok role="UNKNOWN">P</XMTok>
</XMath>
</Math> is capable of
standing on the corners of a triangle <Math mode="inline" tex="T" text="T" xml:id="S1.Thmthm3.p1.m2">
<XMath>
<XMTok role="UNKNOWN">T</XMTok>
</XMath>
</Math> iff (<ref labelref="LABEL:sdq"/>) holds.</text></p>
</para>
</theorem>
</section>
</document>