Simple Polygons of Maximum Perimeter Contained in a Unit Disk

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1 Discrete Comput Geom (009) 1: DOI /s Simple Polygos of Maximum Perimeter Cotaied i a Uit Disk Charles Audet Pierre Hase Frédéric Messie Received: 18 September 007 / Revised: May 008 / Accepted: 6 May 008 / Published olie: 0 Jue 008 Spriger Sciece+Busiess Media, LLC 008 Abstract A polygo is said to be simple if the oly poits of the plae belogig to two of its edges are its vertices. We aswer the questio of fidig, for a give iteger,asimple-sided polygo cotaied i a disk of radius 1 that has the logest perimeter. Whe is eve, the optimal solutio is arbitrarily close to a lie segmet of legth. Whe is odd, the optimal solutio is arbitrarily close to a isosceles triagle. Keywords Simple polygo Perimeter 1 Itroductio I this paper, we give a complete aswer to the questio posed by Brass, Moser ad Pach [1, p. 37]: for a give iteger, fid a simple -sided plaar polygo cotaied i a disk of radius 1 that has the logest perimeter. A polygo is said to be simple, or oitersectig, if the oly poits of the plae belogig to two of its edges are its vertices. These authors metio that the optimal solutio for the case where is eve is arbitrarily close to the trivial upper boud. It suffices to cosider a simple Work of the first author was supported by NSERC grat , AFOSR FA , ad ExxoMobil. Work of the secod author was supported by NSERC grat C. Audet ( ) P. Hase GERAD ad Départemet de Mathématiques et de Géie Idustriel, École Polytechique de Motréal, C.P. 6079, Succ. Cetre-ville, Motréal, Québec, H3C 3A7 Caada Charles.Audet@gerad.ca P. Hase Pierre.Hase@gerad.ca F. Messie ENSEEIHT-IRIT, UMR-CNRS 5505, rue Camichel, BP 71, Toulouse Cedex 7, Frace Frederic.Messie@7.fr

2 Discrete Comput Geom (009) 1: polygo with all vertices with odd idices beig arbitrarily close to ( 1, 0) ad all vertices with eve idices beig arbitrarily close to (1, 0). We show that for = 3, the optimal solutio is the equilateral triagle ad that for odd values of 5, the optimal solutio is arbitrarily close to a isosceles triagle. A Simple Polygo with Logest Perimeter Let 3 be a odd iteger, ad v 1,v,...,v deote the cosecutive vertices of a simple polygo cotaied i a disk of radius 1. Defie N ={1,,...,}, ad for ay i N, letc i = v i+1 v i be the Euclidea distace betwee the two cosecutive vertices v i = (x i,y i ) ad v i+1 = (x i+1,y i+1 ) (throughout this ote, sums of idices are take modulo ). The perimeter of the polygo may the be writte as p = i=1 c i. Defie p to be the supremum of the perimeter i value over all simple polygos cotaied i a disk of radius 1. We use the term supremum istead of maximum sice we will see that there are o optimal simple polygos whe >3. Ideed, we already kow that the optimal solutio for a eve umber of vertices is arbitrarily close to a lie segmet, a figure with oly vertices. For = 3, a equilateral triagle yields a lower boud of p 3 = 3 3op3. For a odd umber of vertices 5, oe ca easily show that p = ( ) + = ( )> 3 is a lower boud o p. To do so, it suffices to costruct a polygo with all of its vertices with eve idex arbitrarily close to (1, 0), the vertex v at (0, 1), ad the remaiig oes arbitrarily close to ( 1, 0). These lower bouds o p are ot tight sice a small displacemet of all vertices at (±1, 0) o the circle (ad away from (0, 1)) produces a ew isosceles triagle. The the derivative of the legth of the edges betwee ( 1, 0) ad (1, 0) is zero, ad the derivative of the legth of the edges betwee (0, 1) ad (±1, 0) is strictly positive. Therefore, a small displacemet will icrease the perimeter, ad we will show that the optimal figure is obtaied by such a displacemet. By relabelig the idices if ecessary, we suppose that ad 1 are the idices of the cosecutive vertices that are the farthest apart, i.e., c c i for all i N. Furthermore, sice we are iterested i maximizig the perimeter, we will cosider that c p. I particular, this implies that c 3 3 ad that c ( )> 3 for odd 5. Furthermore, we will assume that the disk cotaiig the polygo is cetered at (0, 0) ad that the vertices v 1 ad v are vertically aliged: x = x 1 ad y = y 1 + c. Defie U = { (a, b) : a + b 1,b c 1 } ad L = { (a, b) : a + b } 1,b 1 c. The ext result shows that ay two cosecutive vertices of a optimal polygo caot both lie i the same zoe L or U. Lemma 1 If 5 ad if two cosecutive vertices v l ad v l+1 belog to same set U or L, the p<p.

3 10 Discrete Comput Geom (009) 1: Fig. 1 Separatio of the disk ito zoes Proof If v l ad v l+1 areithesamesetl or U (represeted by the shaded regios of Fig. 1), the c l = v l+1 v l 1 (c 1) = c c. It follows that p = i=1 c i ( 1)c + c c. The maximum of the right part of the previous iequality is attaied either whe c is at oe of its boud or 3 or whe the first derivative with respect to c equals zero, i.e., whe ( 1) + (1 c )/ c c = 0. Oe ca easily verify that the maximum occurs whe c = + 1, ad therefore 1 +5 p Furthermore, oe ca show that if 5, the this upper boud o the perimeter is strictly iferior to the lower boud p = + sice + 5 < ( 3 + ). The ext lemma shows that there are three cosecutive vertices that satisfy a orderig property. Lemma There exist three cosecutive vertices v l 1,v l, ad v l+1 for some l {, 3,..., 1} such that either y l 1 y l y l+1 or y l 1 y l y l+1. Proof The previous lemma esures that y > 1 c y 1 ad y 1 <c 1 y.

4 Discrete Comput Geom (009) 1: Fig. Maximizatio of c l 1 + c l without itersectig v 1 v Suppose ow that the result is false. The, y >y 1 implies that y 3 <y. Similarly y 3 <y implies that y >y 3. More geerally, if i is eve, the y i+1 <y i ad y i >y i 1. It follows that, sice is odd, y 1 >y, which is a cotradictio. Without loss of geerality ad by takig vertical ad horizotal symmetrical images if ecessary, we ca assume that the vertex v l of Lemma is such that x l x 1 ad y 1 y > 0 (i.e., that y 1 y 1 + c > 0). Defie u [ 1 c /, 0] to be the smallest scalar such that u + y1 1. The the segmet joiig (u, y 1) ad (u, y 1 + c ) does ot itersect the segmets v l 1 v l ad v l v l+1. These quatities are illustrated o Fig.. The objective of the ext results is to derive a upper boud o the sum c l 1 + c l = v l v l 1 + v l v l+1. The two previous lemmas esure that the vertex v l belogs to the set Ω ={(a, b) : u a 1 b, b 1 c }. Defie the fuctio f : R R as f(x)= dist ( ) ( ) (x, y l ), v l 1 + dist (x, yl ), v l+1 for (x, y l ) Ω. It is a covex fuctio, ad its maximum occurs whe (x, y l ) belogs to the boudary of Ω, i.e., whe x = u or whe x = 1 yl. The ext pair of lemmas cosiders these cases. Lemma 3 If 5 ad v l = (u, y l ), the p<p. Proof If v l = (u, y l ), the c l 1 + c l u + 1 yl + u + yl i Fig. 3. It follows that max u + 1 y + u + y + 1 y s.t. u + y 1 + 1, as illustrated

5 1 Discrete Comput Geom (009) 1: Fig. 3 The vertex v l satisfies x l = u 0 is a valid upper boud o c l 1 + c l. The maximum is attaied whe y = 1 (1 u ), ad the optimal value is z(u) = u + + u + 1. Usig the facts that u 1 c / ad c > 3 whe 5, oe ca show that z(u) + c < 1 3/ + + (1 3/) = 3/ + 5/ + 3. Therefore p ( )c + z(u) = ( 3)c + z(u) + c < ( 3) + 3/ + 5/ + 3 = 9/ + 5/ + 3 < + = p. Lemma If 5 ad v l = ( 1 yl,y l) or if = 3, the c l 1 + c l + 1 c. Proof Cosider the case where v l = ( 1 yl,y l). Defie v 1 = (u, y 1), ad let v = (x,y ) be the poit o the boudary of the disk such that v v 1 =c ad x u. For ay 0 θ π, defie p(θ) to be the poit o the boudary of the circle that makes a agle of θ with v l, as illustrated i the left part of Fig.. The p(0) = v l. Defie 0 θ π θ 1 π to be the agles that satisfy v 1 = p(θ 1 ) ad v = p(θ ).

6 Discrete Comput Geom (009) 1: Fig. The poit o the disk at maximum distace from ( c, 0) ad ( c, 0) The distace fuctio, p(θ) v l is icreasig for θ [0,π] ad decreasig for θ [π,π]. However, if θ <θ<θ 1, the the segmet joiig p(θ) ad v l itersects the segmet v 1 v. It follows that the sum c l 1 + c l is bouded above by v l v + v l v 1. To simplify the presetatio, let us move the disk so that the segmet v 1 v is horizotal with ed poits (±c /, 0), ad the coordiates of v l become (a, b) with b 0, as illustrated i the right part of Fig.. Observe that the case where = 3is represeted by the right side of that figure, as all three vertices of the optimal triagle ecessarily lie o the boudary of the disk. Equatio (1) provides a upper boud o the sum: ( ) ( ) (a, b) + c, 0 + (a, b) c, 0 max p + q a,b,p,q ( s.t. p = a + c ) + b, ( q = a c ) (1) + b, ( a + b 1 c ) = 1. The Lagrage multipliers system is ( 0 = a + c ) ( λ 1 + a c ) λ + aλ 3, ( ) 0 = bλ 1 + bλ + b 1 c λ 3,

7 1 Discrete Comput Geom (009) 1: = pλ 1, 1 = qλ. Solvig the two last equatios for the Lagrage multipliers ad substitutig them ito the two first oes gives ( q a + c ) ( + p a c ) = apqλ 3, ( ) (q + p)b = pq b 1 c After elimiatig λ 3 ad makig some simplificatios, oe gets ( ) (q p) c b 1 c = (q + p)a 1 c. () Multiplyig both sides by (q p) ad usig the costraits of (1) to substitute (q + p)(q p) = q p = ac leads to ( ) (q p) c b 1 c = a c 1 c. Recall that we are lookig for a solutio with b 1 c 0 ad c > 0. Therefore, the left-had-side term is oegative, ad the right-had-side term is opositive. To get a equality, we eed that a = 0 or that a 0 ad c =. If a = 0, the p = q = + b = c c + ( c λ 3. ) = + 1 c. If a 0 ad c =, the () becomes b(p q) = 0. The case where b = 0 correspods to a miimizer, ad p = q implies that (a + 1) = (a 1), i.e., that a = 0, which is a cotradictio. Therefore, we have show that c l 1 + c l + 1 c. Combiig the above lemmas leads to our mai result. Theorem 1 If is a odd iteger, the the supremum of the perimeter i value over all simple polygos cotaied i a disk of radius oe is p = ( 1 + 8( ) 1) 1 ( 1 + 8( ) + 3) 3 ( ) ad the polygo is arbitrarily close to a isosceles triagle.,

8 Discrete Comput Geom (009) 1: Proof Combiig Lemmas 3 ad ad settig v = 1 c leads to p ( )c c = ( ) 1 v + + v. I order to miimize this upper boud, we take the derivative ad set it equal to zero. This gives v ( ) + = 0 1 v 1 + v 1 + v ad may be rewritte as ( ) v + v 1 = 0. The uique oegative solutio is v = ( ). Substitutig for v ad simplifyig leads to ( ) p ( 1 + 8( ) 1) 1 ( 1 + 8( ) + 3) 3. ( ) A simple polygo with a perimeter arbitrarily close to that value ca be costructed. Cosider a simple polygo i which ( 1) cosecutive vertices alterate back ad forth arbitrarily close to ( c, 0) ad ( c, 0), ad the remaiig vertex is located at (0, 1 + v), where c = 1 v ad v = ( ). The resultig ( ) figure is arbitrarily close to a isosceles triagle, ad i the limit, the perimeter satisfies p = ( 1 + 8( ) 1) 1 ( 1 + 8( ) + 3) 3. ( ) I the case where = 3, the optimal triagle is equilateral. We coclude with the followig corollary. Corollary 1 For odd values of, the sequece { p } is mootoe icreasig. Proof For ay fixed value of, maximizig the perimeter or maximizig the average edge legth yields the same figure. Furthermore, for ay fixed odd value of, oe ca easily costruct a ( + )-sided polygo whose average edge legth strictly exceeds p. Ideed, oe simply has to cosider a optimal -sided polygo ad add a vertex ear ( a,0) ad aother oe ear (a, 0). This will add two edges of legth arbitrarily close to the value c > p. Refereces 1. Brass, P., Moser, W., Pach, J.: Research Problems i Discrete Geometry. Spriger, New York (005)

Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions

Math 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x