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|00001120| 7b 4c 65 6d 6d 61 7d 0a | 0a 25 20 20 20 20 42 65 |{Lemma}.|.% Be|
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|000011a0| 72 74 3d 7c 5c 6c 65 74 | 5c 72 76 65 72 74 3d 7c |rt=|\let|\rvert=||
|000011b0| 0a 5c 6e 65 77 63 6f 6d | 6d 61 6e 64 7b 5c 52 69 |.\newcom|mand{\Ri|
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|00001340| 66 7b 44 7d 5f 72 24 2c | 20 24 68 5c 69 6e 20 43 |f{D}_r$,| $h\in C|
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|00001360| 24 2c 20 77 69 74 68 20 | 24 5c 6f 6d 65 67 61 24 |$, with |$\omega$|
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|00001380| 24 28 31 2c 31 29 24 2d | 66 6f 72 6d 2e 20 49 66 |$(1,1)$-|form. If|
|00001390| 20 24 5c 52 69 63 5c 6f | 6d 65 67 61 5c 67 65 71 | $\Ric\o|mega\geq|
|000013a0| 5c 6f 6d 65 67 61 24 20 | 6f 6e 20 24 5c 6d 61 74 |\omega$ |on $\mat|
|000013b0| 68 62 66 7b 44 7d 5f 72 | 24 2c 0a 74 68 65 6e 20 |hbf{D}_r|$,.then |
|000013c0| 24 5c 6f 6d 65 67 61 5c | 6c 65 71 5c 6f 6d 65 67 |$\omega\|leq\omeg|
|000013d0| 61 5f 72 24 20 6f 6e 20 | 61 6c 6c 20 6f 66 20 24 |a_r$ on |all of $|
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|00001400| 24 64 73 5e 32 5c 6c 65 | 71 20 64 73 5f 72 5e 32 |$ds^2\le|q ds_r^2|
|00001410| 24 29 2e 0a 5c 65 6e 64 | 7b 41 68 6c 66 6f 72 73 |$)..\end|{Ahlfors|
|00001420| 7d 0a 0a 5c 62 65 67 69 | 6e 7b 6c 65 6d 7d 5b 6e |}..\begi|n{lem}[n|
|00001430| 65 67 61 74 69 76 65 6c | 79 20 63 75 72 76 65 64 |egativel|y curved|
|00001440| 20 66 61 6d 69 6c 69 65 | 73 5d 0a 4c 65 74 20 24 | familie|s].Let $|
|00001450| 5c 7b 64 73 5f 31 5e 32 | 2c 5c 64 6f 74 73 2c 64 |\{ds_1^2|,\dots,d|
|00001460| 73 5f 6b 5e 32 5c 7d 24 | 20 62 65 20 61 20 6e 65 |s_k^2\}$| be a ne|
|00001470| 67 61 74 69 76 65 6c 79 | 20 63 75 72 76 65 64 20 |gatively| curved |
|00001480| 66 61 6d 69 6c 79 20 6f | 66 20 6d 65 74 72 69 63 |family o|f metric|
|00001490| 73 0a 6f 6e 20 24 5c 6d | 61 74 68 62 66 7b 44 7d |s.on $\m|athbf{D}|
|000014a0| 5f 72 24 2c 20 77 69 74 | 68 20 61 73 73 6f 63 69 |_r$, wit|h associ|
|000014b0| 61 74 65 64 20 66 6f 72 | 6d 73 20 24 5c 6f 6d 65 |ated for|ms $\ome|
|000014c0| 67 61 5e 31 24 2c 20 5c | 64 6f 74 73 2c 20 24 5c |ga^1$, \|dots, $\|
|000014d0| 6f 6d 65 67 61 5e 6b 24 | 2e 0a 54 68 65 6e 20 24 |omega^k$|..Then $|
|000014e0| 5c 6f 6d 65 67 61 5e 69 | 20 5c 6c 65 71 5c 6f 6d |\omega^i| \leq\om|
|000014f0| 65 67 61 5f 72 24 20 66 | 6f 72 20 61 6c 6c 20 24 |ega_r$ f|or all $|
|00001500| 69 24 2e 0a 5c 65 6e 64 | 7b 6c 65 6d 7d 0a 0a 54 |i$..\end|{lem}..T|
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|00001520| 6f 72 65 6d 3a 0a 5c 62 | 65 67 69 6e 7b 74 68 6d |orem:.\b|egin{thm|
|00001530| 7d 5c 6c 61 62 65 6c 7b | 70 69 67 73 70 61 6e 7d |}\label{|pigspan}|
|00001540| 0a 4c 65 74 20 24 64 5f | 7b 5c 6d 61 78 7d 24 20 |.Let $d_|{\max}$ |
|00001550| 61 6e 64 20 24 64 5f 7b | 5c 6d 69 6e 7d 24 20 62 |and $d_{|\min}$ b|
|00001560| 65 20 74 68 65 20 6d 61 | 78 69 6d 75 6d 2c 20 72 |e the ma|ximum, r|
|00001570| 65 73 70 2e 5c 20 6d 69 | 6e 69 6d 75 6d 20 64 69 |esp.\ mi|nimum di|
|00001580| 73 74 61 6e 63 65 0a 62 | 65 74 77 65 65 6e 20 61 |stance.b|etween a|
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|000015a0| 76 65 72 74 69 63 65 73 | 20 6f 66 20 61 20 71 75 |vertices| of a qu|
|000015b0| 61 64 72 69 6c 61 74 65 | 72 61 6c 20 24 51 24 2e |adrilate|ral $Q$.|
|000015c0| 20 4c 65 74 20 24 5c 73 | 69 67 6d 61 24 0a 62 65 | Let $\s|igma$.be|
|000015d0| 20 74 68 65 20 64 69 61 | 67 6f 6e 61 6c 20 70 69 | the dia|gonal pi|
|000015e0| 67 73 70 61 6e 20 6f 66 | 20 61 20 70 69 67 20 24 |gspan of| a pig $|
|000015f0| 50 24 20 77 69 74 68 20 | 66 6f 75 72 20 6c 65 67 |P$ with |four leg|
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|00001620| 6e 67 20 6f 6e 20 74 68 | 65 20 63 6f 72 6e 65 72 |ng on th|e corner|
|00001630| 73 20 6f 66 20 24 51 24 | 20 69 66 66 0a 5c 62 65 |s of $Q$| iff.\be|
|00001640| 67 69 6e 7b 65 71 75 61 | 74 69 6f 6e 7d 5c 6c 61 |gin{equa|tion}\la|
|00001650| 62 65 6c 7b 73 64 71 7d | 0a 5c 73 69 67 6d 61 5c |bel{sdq}|.\sigma\|
|00001660| 67 65 71 20 5c 73 71 72 | 74 7b 64 5f 7b 5c 6d 61 |geq \sqr|t{d_{\ma|
|00001670| 78 7d 5e 32 2b 64 5f 7b | 5c 6d 69 6e 7d 5e 32 7d |x}^2+d_{|\min}^2}|
|00001680| 2e 0a 5c 65 6e 64 7b 65 | 71 75 61 74 69 6f 6e 7d |..\end{e|quation}|
|00001690| 0a 5c 65 6e 64 7b 74 68 | 6d 7d 0a 0a 5c 62 65 67 |.\end{th|m}..\beg|
|000016a0| 69 6e 7b 63 6f 72 7d 0a | 41 64 6d 69 74 74 69 6e |in{cor}.|Admittin|
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|000016c0| 20 72 6f 74 61 74 69 6f | 6e 2c 20 61 20 74 68 72 | rotatio|n, a thr|
|000016d0| 65 65 2d 6c 65 67 67 65 | 64 20 70 69 67 20 24 50 |ee-legge|d pig $P|
|000016e0| 24 20 69 73 20 63 61 70 | 61 62 6c 65 20 6f 66 0a |$ is cap|able of.|
|000016f0| 73 74 61 6e 64 69 6e 67 | 20 6f 6e 20 74 68 65 20 |standing| on the |
|00001700| 63 6f 72 6e 65 72 73 20 | 6f 66 20 61 20 74 72 69 |corners |of a tri|
|00001710| 61 6e 67 6c 65 20 24 54 | 24 20 69 66 66 20 28 5c |angle $T|$ iff (\|
|00001720| 72 65 66 7b 73 64 71 7d | 29 20 68 6f 6c 64 73 2e |ref{sdq}|) holds.|
|00001730| 0a 5c 65 6e 64 7b 63 6f | 72 7d 0a 0a 5c 62 65 67 |.\end{co|r}..\beg|
|00001740| 69 6e 7b 72 6d 6b 7d 0a | 41 73 20 74 77 6f 2d 6c |in{rmk}.|As two-l|
|00001750| 65 67 67 65 64 20 70 69 | 67 73 20 67 65 6e 65 72 |egged pi|gs gener|
|00001760| 61 6c 6c 79 20 66 61 6c | 6c 20 6f 76 65 72 2c 20 |ally fal|l over, |
|00001770| 74 68 65 20 63 61 73 65 | 20 6f 66 20 61 20 70 6f |the case| of a po|
|00001780| 6c 79 67 6f 6e 20 6f 66 | 20 6f 72 64 65 72 0a 24 |lygon of| order.$|
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|000017a0| 69 6e 67 2e 0a 5c 65 6e | 64 7b 72 6d 6b 7d 0a 0a |ing..\en|d{rmk}..|
|000017b0| 5c 62 65 67 69 6e 7b 65 | 78 65 72 7d 0a 47 65 6e |\begin{e|xer}.Gen|
|000017c0| 65 72 61 6c 69 7a 65 20 | 54 68 65 6f 72 65 6d 7e |eralize |Theorem~|
|000017d0| 5c 72 65 66 7b 70 69 67 | 73 70 61 6e 7d 20 74 6f |\ref{pig|span} to|
|000017e0| 20 74 68 72 65 65 20 61 | 6e 64 20 66 6f 75 72 20 | three a|nd four |
|000017f0| 64 69 6d 65 6e 73 69 6f | 6e 73 2e 0a 5c 65 6e 64 |dimensio|ns..\end|
|00001800| 7b 65 78 65 72 7d 0a 0a | 5c 62 65 67 69 6e 7b 6e |{exer}..|\begin{n|
|00001810| 6f 74 65 7d 0a 54 68 69 | 73 20 69 73 20 61 20 74 |ote}.Thi|s is a t|
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|00001830| 6d 20 74 68 65 6f 72 65 | 6d 20 73 74 79 6c 65 20 |m theore|m style |
|00001840| 60 6e 6f 74 65 27 2e 20 | 49 74 20 69 73 20 73 75 |`note'. |It is su|
|00001850| 70 70 6f 73 65 64 20 74 | 6f 20 68 61 76 65 0a 76 |pposed t|o have.v|
|00001860| 61 72 69 61 6e 74 20 66 | 6f 6e 74 73 20 61 6e 64 |ariant f|onts and|
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|00001880| 65 73 2e 0a 5c 65 6e 64 | 7b 6e 6f 74 65 7d 0a 0a |es..\end|{note}..|
|00001890| 5c 62 65 67 69 6e 7b 62 | 74 68 6d 7d 0a 54 65 73 |\begin{b|thm}.Tes|
|000018a0| 74 20 6f 66 20 74 68 65 | 20 60 6c 69 6e 65 62 72 |t of the| `linebr|
|000018b0| 65 61 6b 27 20 73 74 79 | 6c 65 20 6f 66 20 74 68 |eak' sty|le of th|
|000018c0| 65 6f 72 65 6d 20 68 65 | 61 64 69 6e 67 2e 0a 5c |eorem he|ading..\|
|000018d0| 65 6e 64 7b 62 74 68 6d | 7d 0a 0a 54 68 69 73 20 |end{bthm|}..This |
|000018e0| 69 73 20 61 20 74 65 73 | 74 20 6f 66 20 61 20 63 |is a tes|t of a c|
|000018f0| 69 74 69 6e 67 20 74 68 | 65 6f 72 65 6d 20 74 6f |iting th|eorem to|
|00001900| 20 63 69 74 65 20 61 20 | 74 68 65 6f 72 65 6d 20 | cite a |theorem |
|00001910| 66 72 6f 6d 20 73 6f 6d | 65 20 6f 74 68 65 72 20 |from som|e other |
|00001920| 73 6f 75 72 63 65 2e 0a | 0a 5c 62 65 67 69 6e 7b |source..|.\begin{|
|00001930| 76 61 72 74 68 6d 7d 5b | 54 68 65 6f 72 65 6d 20 |varthm}[|Theorem |
|00001940| 33 2e 36 20 69 6e 20 5c | 63 69 74 65 7b 74 68 61 |3.6 in \|cite{tha|
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|00001960| 69 6e 6b 69 6e 67 20 61 | 76 61 69 6c 61 62 6c 65 |inking a|vailable|
|00001970| 20 68 65 72 65 20 79 65 | 74 20 5c 64 6f 74 73 5c | here ye|t \dots\|
|00001980| 20 62 75 74 20 74 68 61 | 74 27 73 20 6e 6f 74 20 | but tha|t's not |
|00001990| 61 0a 62 61 64 20 69 64 | 65 61 20 66 6f 72 20 74 |a.bad id|ea for t|
|000019a0| 68 65 20 66 75 74 75 72 | 65 2e 0a 5c 65 6e 64 7b |he futur|e..\end{|
|000019b0| 76 61 72 74 68 6d 7d 0a | 0a 5c 62 65 67 69 6e 7b |varthm}.|.\begin{|
|000019c0| 70 72 6f 6f 66 7d 0a 48 | 65 72 65 20 69 73 20 61 |proof}.H|ere is a|
|000019d0| 20 74 65 73 74 20 6f 66 | 20 74 68 65 20 70 72 6f | test of| the pro|
|000019e0| 6f 66 20 65 6e 76 69 72 | 6f 6e 6d 65 6e 74 2e 0a |of envir|onment..|
|000019f0| 5c 65 6e 64 7b 70 72 6f | 6f 66 7d 0a 0a 5c 62 65 |\end{pro|of}..\be|
|00001a00| 67 69 6e 7b 70 72 6f 6f | 66 7d 5b 50 72 6f 6f 66 |gin{proo|f}[Proof|
|00001a10| 20 6f 66 20 54 68 65 6f | 72 65 6d 20 5c 72 65 66 | of Theo|rem \ref|
|00001a20| 7b 70 69 67 73 70 61 6e | 7d 5d 0a 41 6e 64 20 61 |{pigspan|}].And a|
|00001a30| 6e 6f 74 68 65 72 20 74 | 65 73 74 2e 0a 5c 65 6e |nother t|est..\en|
|00001a40| 64 7b 70 72 6f 6f 66 7d | 0a 0a 5c 62 65 67 69 6e |d{proof}|..\begin|
|00001a50| 7b 70 72 6f 6f 66 7d 5b | 50 72 6f 6f 66 20 28 6e |{proof}[|Proof (n|
|00001a60| 65 63 65 73 73 69 74 79 | 29 5d 0a 41 6e 64 20 61 |ecessity|)].And a|
|00001a70| 6e 6f 74 68 65 72 2e 0a | 5c 65 6e 64 7b 70 72 6f |nother..|\end{pro|
|00001a80| 6f 66 7d 0a 0a 5c 62 65 | 67 69 6e 7b 70 72 6f 6f |of}..\be|gin{proo|
|00001a90| 66 7d 5b 50 72 6f 6f 66 | 20 28 73 75 66 66 69 63 |f}[Proof| (suffic|
|00001aa0| 69 65 6e 63 79 29 5d 0a | 41 6e 64 20 61 6e 6f 74 |iency)].|And anot|
|00001ab0| 68 65 72 2e 0a 5c 65 6e | 64 7b 70 72 6f 6f 66 7d |her..\en|d{proof}|
|00001ac0| 0a 0a 5c 73 65 63 74 69 | 6f 6e 7b 54 65 73 74 20 |..\secti|on{Test |
|00001ad0| 6f 66 20 6e 75 6d 62 65 | 72 2d 73 77 61 70 70 69 |of numbe|r-swappi|
|00001ae0| 6e 67 7d 0a 0a 54 68 69 | 73 20 69 73 20 61 20 72 |ng}..Thi|s is a r|
|00001af0| 65 70 65 61 74 20 6f 66 | 20 74 68 65 20 66 69 72 |epeat of| the fir|
|00001b00| 73 74 20 73 65 63 74 69 | 6f 6e 20 62 75 74 20 77 |st secti|on but w|
|00001b10| 69 74 68 20 6e 75 6d 62 | 65 72 73 20 69 6e 20 74 |ith numb|ers in t|
|00001b20| 68 65 6f 72 65 6d 20 68 | 65 61 64 73 0a 73 77 61 |heorem h|eads.swa|
|00001b30| 70 70 65 64 20 74 6f 20 | 74 68 65 20 6c 65 66 74 |pped to |the left|
|00001b40| 2e 0a 0a 41 68 6c 66 6f | 72 73 27 20 4c 65 6d 6d |...Ahlfo|rs' Lemm|
|00001b50| 61 20 67 69 76 65 73 20 | 74 68 65 20 70 72 69 6e |a gives |the prin|
|00001b60| 63 69 70 61 6c 20 63 72 | 69 74 65 72 69 6f 6e 20 |cipal cr|iterion |
|00001b70| 66 6f 72 20 6f 62 74 61 | 69 6e 69 6e 67 20 6c 6f |for obta|ining lo|
|00001b80| 77 65 72 20 62 6f 75 6e | 64 73 0a 6f 6e 20 74 68 |wer boun|ds.on th|
|00001b90| 65 20 4b 6f 62 61 79 61 | 73 68 69 20 6d 65 74 72 |e Kobaya|shi metr|
|00001ba0| 69 63 2e 0a 5c 62 65 67 | 69 6e 7b 41 68 6c 66 6f |ic..\beg|in{Ahlfo|
|00001bb0| 72 73 7d 0a 4c 65 74 20 | 24 64 73 5e 32 20 3d 20 |rs}.Let |$ds^2 = |
|00001bc0| 68 28 7a 29 5c 6c 76 65 | 72 74 20 64 7a 5c 72 76 |h(z)\lve|rt dz\rv|
|00001bd0| 65 72 74 5e 32 24 20 62 | 65 20 61 20 48 65 72 6d |ert^2$ b|e a Herm|
|00001be0| 69 74 69 61 6e 20 70 73 | 65 75 64 6f 2d 6d 65 74 |itian ps|eudo-met|
|00001bf0| 72 69 63 20 6f 6e 0a 24 | 5c 6d 61 74 68 62 66 7b |ric on.$|\mathbf{|
|00001c00| 44 7d 5f 72 24 2c 20 24 | 68 5c 69 6e 20 43 5e 32 |D}_r$, $|h\in C^2|
|00001c10| 28 5c 6d 61 74 68 62 66 | 7b 44 7d 5f 72 29 24 2c |(\mathbf|{D}_r)$,|
|00001c20| 20 77 69 74 68 20 24 5c | 6f 6d 65 67 61 24 20 74 | with $\|omega$ t|
|00001c30| 68 65 20 61 73 73 6f 63 | 69 61 74 65 64 0a 24 28 |he assoc|iated.$(|
|00001c40| 31 2c 31 29 24 2d 66 6f | 72 6d 2e 20 49 66 20 24 |1,1)$-fo|rm. If $|
|00001c50| 5c 52 69 63 5c 6f 6d 65 | 67 61 5c 67 65 71 5c 6f |\Ric\ome|ga\geq\o|
|00001c60| 6d 65 67 61 24 20 6f 6e | 20 24 5c 6d 61 74 68 62 |mega$ on| $\mathb|
|00001c70| 66 7b 44 7d 5f 72 24 2c | 0a 74 68 65 6e 20 24 5c |f{D}_r$,|.then $\|
|00001c80| 6f 6d 65 67 61 5c 6c 65 | 71 5c 6f 6d 65 67 61 5f |omega\le|q\omega_|
|00001c90| 72 24 20 6f 6e 20 61 6c | 6c 20 6f 66 20 24 5c 6d |r$ on al|l of $\m|
|00001ca0| 61 74 68 62 66 7b 44 7d | 5f 72 24 20 28 6f 72 20 |athbf{D}|_r$ (or |
|00001cb0| 65 71 75 69 76 61 6c 65 | 6e 74 6c 79 2c 0a 24 64 |equivale|ntly,.$d|
|00001cc0| 73 5e 32 5c 6c 65 71 20 | 64 73 5f 72 5e 32 24 29 |s^2\leq |ds_r^2$)|
|00001cd0| 2e 0a 5c 65 6e 64 7b 41 | 68 6c 66 6f 72 73 7d 0a |..\end{A|hlfors}.|
|00001ce0| 0a 5c 62 65 67 69 6e 7b | 6c 65 6d 73 77 7d 5b 6e |.\begin{|lemsw}[n|
|00001cf0| 65 67 61 74 69 76 65 6c | 79 20 63 75 72 76 65 64 |egativel|y curved|
|00001d00| 20 66 61 6d 69 6c 69 65 | 73 5d 0a 4c 65 74 20 24 | familie|s].Let $|
|00001d10| 5c 7b 64 73 5f 31 5e 32 | 2c 5c 64 6f 74 73 2c 64 |\{ds_1^2|,\dots,d|
|00001d20| 73 5f 6b 5e 32 5c 7d 24 | 20 62 65 20 61 20 6e 65 |s_k^2\}$| be a ne|
|00001d30| 67 61 74 69 76 65 6c 79 | 20 63 75 72 76 65 64 20 |gatively| curved |
|00001d40| 66 61 6d 69 6c 79 20 6f | 66 20 6d 65 74 72 69 63 |family o|f metric|
|00001d50| 73 0a 6f 6e 20 24 5c 6d | 61 74 68 62 66 7b 44 7d |s.on $\m|athbf{D}|
|00001d60| 5f 72 24 2c 20 77 69 74 | 68 20 61 73 73 6f 63 69 |_r$, wit|h associ|
|00001d70| 61 74 65 64 20 66 6f 72 | 6d 73 20 24 5c 6f 6d 65 |ated for|ms $\ome|
|00001d80| 67 61 5e 31 24 2c 20 5c | 64 6f 74 73 2c 20 24 5c |ga^1$, \|dots, $\|
|00001d90| 6f 6d 65 67 61 5e 6b 24 | 2e 0a 54 68 65 6e 20 24 |omega^k$|..Then $|
|00001da0| 5c 6f 6d 65 67 61 5e 69 | 20 5c 6c 65 71 5c 6f 6d |\omega^i| \leq\om|
|00001db0| 65 67 61 5f 72 24 20 66 | 6f 72 20 61 6c 6c 20 24 |ega_r$ f|or all $|
|00001dc0| 69 24 2e 0a 5c 65 6e 64 | 7b 6c 65 6d 73 77 7d 0a |i$..\end|{lemsw}.|
|00001dd0| 0a 54 68 65 6e 20 6f 75 | 72 20 6d 61 69 6e 20 74 |.Then ou|r main t|
|00001de0| 68 65 6f 72 65 6d 3a 0a | 5c 62 65 67 69 6e 7b 74 |heorem:.|\begin{t|
|00001df0| 68 6d 73 77 7d 0a 4c 65 | 74 20 24 64 5f 7b 5c 6d |hmsw}.Le|t $d_{\m|
|00001e00| 61 78 7d 24 20 61 6e 64 | 20 24 64 5f 7b 5c 6d 69 |ax}$ and| $d_{\mi|
|00001e10| 6e 7d 24 20 62 65 20 74 | 68 65 20 6d 61 78 69 6d |n}$ be t|he maxim|
|00001e20| 75 6d 2c 20 72 65 73 70 | 2e 5c 20 6d 69 6e 69 6d |um, resp|.\ minim|
|00001e30| 75 6d 20 64 69 73 74 61 | 6e 63 65 0a 62 65 74 77 |um dista|nce.betw|
|00001e40| 65 65 6e 20 61 6e 79 20 | 74 77 6f 20 61 64 6a 61 |een any |two adja|
|00001e50| 63 65 6e 74 20 76 65 72 | 74 69 63 65 73 20 6f 66 |cent ver|tices of|
|00001e60| 20 61 20 71 75 61 64 72 | 69 6c 61 74 65 72 61 6c | a quadr|ilateral|
|00001e70| 20 24 51 24 2e 20 4c 65 | 74 20 24 5c 73 69 67 6d | $Q$. Le|t $\sigm|
|00001e80| 61 24 0a 62 65 20 74 68 | 65 20 64 69 61 67 6f 6e |a$.be th|e diagon|
|00001e90| 61 6c 20 70 69 67 73 70 | 61 6e 20 6f 66 20 61 20 |al pigsp|an of a |
|00001ea0| 70 69 67 20 24 50 24 20 | 77 69 74 68 20 66 6f 75 |pig $P$ |with fou|
|00001eb0| 72 20 6c 65 67 73 2e 0a | 54 68 65 6e 20 24 50 24 |r legs..|Then $P$|
|00001ec0| 20 69 73 20 63 61 70 61 | 62 6c 65 20 6f 66 20 73 | is capa|ble of s|
|00001ed0| 74 61 6e 64 69 6e 67 20 | 6f 6e 20 74 68 65 20 63 |tanding |on the c|
|00001ee0| 6f 72 6e 65 72 73 20 6f | 66 20 24 51 24 20 69 66 |orners o|f $Q$ if|
|00001ef0| 66 0a 5c 62 65 67 69 6e | 7b 65 71 75 61 74 69 6f |f.\begin|{equatio|
|00001f00| 6e 7d 5c 6c 61 62 65 6c | 7b 73 64 71 73 77 7d 0a |n}\label|{sdqsw}.|
|00001f10| 5c 73 69 67 6d 61 5c 67 | 65 71 20 5c 73 71 72 74 |\sigma\g|eq \sqrt|
|00001f20| 7b 64 5f 7b 5c 6d 61 78 | 7d 5e 32 2b 64 5f 7b 5c |{d_{\max|}^2+d_{\|
|00001f30| 6d 69 6e 7d 5e 32 7d 2e | 0a 5c 65 6e 64 7b 65 71 |min}^2}.|.\end{eq|
|00001f40| 75 61 74 69 6f 6e 7d 0a | 5c 65 6e 64 7b 74 68 6d |uation}.|\end{thm|
|00001f50| 73 77 7d 0a 0a 5c 62 65 | 67 69 6e 7b 63 6f 72 73 |sw}..\be|gin{cors|
|00001f60| 77 7d 0a 41 64 6d 69 74 | 74 69 6e 67 20 72 65 66 |w}.Admit|ting ref|
|00001f70| 6c 65 63 74 69 6f 6e 20 | 61 6e 64 20 72 6f 74 61 |lection |and rota|
|00001f80| 74 69 6f 6e 2c 20 61 20 | 74 68 72 65 65 2d 6c 65 |tion, a |three-le|
|00001f90| 67 67 65 64 20 70 69 67 | 20 24 50 24 20 69 73 20 |gged pig| $P$ is |
|00001fa0| 63 61 70 61 62 6c 65 20 | 6f 66 0a 73 74 61 6e 64 |capable |of.stand|
|00001fb0| 69 6e 67 20 6f 6e 20 74 | 68 65 20 63 6f 72 6e 65 |ing on t|he corne|
|00001fc0| 72 73 20 6f 66 20 61 20 | 74 72 69 61 6e 67 6c 65 |rs of a |triangle|
|00001fd0| 20 24 54 24 20 69 66 66 | 20 28 5c 72 65 66 7b 73 | $T$ iff| (\ref{s|
|00001fe0| 64 71 73 77 7d 29 20 68 | 6f 6c 64 73 2e 0a 5c 65 |dqsw}) h|olds..\e|
|00001ff0| 6e 64 7b 63 6f 72 73 77 | 7d 0a 0a 5c 62 65 67 69 |nd{corsw|}..\begi|
|00002000| 6e 7b 74 68 65 62 69 62 | 6c 69 6f 67 72 61 70 68 |n{thebib|liograph|
|00002010| 79 7d 7b 39 39 7d 0a 5c | 62 69 62 69 74 65 6d 7b |y}{99}.\|bibitem{|
|00002020| 74 68 61 74 6f 6e 65 7d | 20 44 75 6d 6d 79 20 65 |thatone}| Dummy e|
|00002030| 6e 74 72 79 2e 0a 5c 65 | 6e 64 7b 74 68 65 62 69 |ntry..\e|nd{thebi|
|00002040| 62 6c 69 6f 67 72 61 70 | 68 79 7d 0a 0a 5c 65 6e |bliograp|hy}..\en|
|00002050| 64 7b 64 6f 63 75 6d 65 | 6e 74 7d 0a |d{docume|nt}. |
+--------+-------------------------+-------------------------+--------+--------+
