The discrete square peg problem

THE DISCRETE S QUARE PEG PR OBLEM IGOR P AK ⋆ Abstract. The square p eg pro ble m asks whether every Jorda n cur ve in the plane has four po in ts whic h form a square. The problem has b een resolved ( po sitiv ely) for v a rious classes of curves, but remains o pen in full g enerality . W e present t w o new direct proo fs for the case of piecewise linear curv es. Intr oduction The squar e p e g pr oble m is beautiful and deceptiv ely simple. It asks whether ev ery Jordan curv e C ⊂ R 2 has fo ur p oints whic h fo rm a square. W e call suc h squares inscrib e d into C (se e Figure 1). P S f r a g r e p la c e m e n t s C Figure 1. Jordan curv e C and an inscrib ed square. The pro blem go es back to T o eplitz (1 911), and ov er almost a cen tury has b een rep eatedly redisco vered and in v estigated, but nev er completely resolv ed. By no w it has been established for con vex c urves and curv es with v arious regularit y c onditions, including the case of piecewise linear curve s. While there are sev eral simple and elegan t pro ofs of the con v ex case, the piecewise linear case is usually o btained as a consequenc e of results pro v ed by ra t her tec hnical top ological and analytic argumen ts. In fact, until to this pap er, there w as no direct elemen tary pro of. Here w e presen t t wo suc h pro ofs in the piecewise linear case. Main Theorem. Every simple p olygon on a plane has an i n scrib e d squar e. As the reader will see, b oth pro ofs are direct a nd elem entary , although p erhaps not to the extend one would call them “b o ok pro ofs”. The pro ofs a re strongly mo t iv ated Date : April 3, 2008. ⋆ School of Mathematics, Univ ersity of Minnesota, Minneapolis, MN, 55455 . E mail: pak @umn.edu . 1 2 IGOR P A K b y the classic a l ideas in the field (see Section 3). Here and there, w e omit a n um b er of minor straigh tfo rw ard details, in particular the deformatio n construction at the end of the second pro of. The rest of the pap er is structured as f ollo ws. In the next tw o sections w e presen t the pro ofs of the main theorem. These pro ofs are comple tely separate a nd can b e read indep enden tly . In the last section w e giv e an outline of the ric h history of the problem and the underlying ideas. The historical part is not meant to b e comprehensiv e, but w e do include a n um b er of p oin t ers to surv eys and recen t references. 1. Proof via insc ribed triangles Let X = [ x 1 . . . x n ] ∈ R 2 b e a simple p olygon. W e a ssume that X is generic in a sense whic h will b e clear later on. F urther, w e assume that the angles of X are obtuse , i.e. lie b etw een π / 2 and 3 π / 2. Fix a clo c kwise or ientation on X . F or an ordered pair ( y , z ) of p oin ts y , z ∈ X denote b y u and v the other t w o v ertices of a square [ z y uv ] in the plane, with ve r t ices on X in t his order, as sho wn in Figure 2. P a r a meterize X b y the length and think of ( y , z ) as a p oin t on a t o rus T = X × X . Denote b y U ⊂ T the subset of pairs ( y , z ) so tha t u ∈ X . Similarly , denote by V ⊂ T the subset of pairs ( y , z ) so that v ∈ X . Our goa l is to sho w that U in tersects V . P S f r a g r e p la c e m e n t s y y z z u u v v X X X ′ Figure 2. Square [ z y uv ] inscrib ed into X and a rota tion X ′ of X a round y . First, observ e that for a generic X the set of p oints U y = { z : ( y , z ) ∈ U } is finite. Indeed, these p oin ts z ∈ U y lie in the in tersection of t he p olygon X and a p olygon X ′ obtained by a coun terclo c kwise rotation of X around y b y an angle π / 2 (see F ig ure 2 ). Therefore, if X do es not ha v e orthogo nal edges there is only a finite n um b er of p oin ts in X ∩ X ′ . Moreo v er, when y mo ves along X a t a constan t sp eed, t hese in tersection p oin ts z change piecewise linearly , whic h implies that U is also piecewise linear. Let us sho w t ha t U is a disjoint union o f simple p olygons. Observ e that when y is mo ve d along X the inters ection p o int z ∈ X ∩ X ′ cannot disapp ear except when a v ertex of X passes throug h an edge of X ′ , or when a v ertex of X ′ passes thr o ugh an edge of X . This implies that when y is mo ved along X the in tersection p oin ts emerge and disapp ear in pairs, and th us U is a union of p olygons. Note that for a generic X , at no time can a v ertex pass through a verte x, whic h is equiv alen t to the conditio n that no square with a dia g onal ( x i , x j ) can hav e a p oin t y ∈ X as its third v ertex. THE DISCRETE SQ UARE PEG PROBLEM 3 T o see that the p olygons in U are simple and disjoint, observ e t hat the o nly w ay w e can ha v e an in tersection if a v ertex of X ′ c hanges direction at a n edge in X , or, similarly , if a v ertex o f X changes direction at an edge in X ′ . This is p ossible only when y and either z or u ar e vertices of X . Since X is c hosen to b e generic w e can assume this do es not ha ppen, i.e. tha t X do es not ha ve an inscrib ed right isosceles triangle with an edge ( x i , x j ). P S f r a g r e p la c e m e n t s X X X X X ′ X ′ X ′ Figure 3. Tw o disapp earing p o in ts in X ∩ X ′ and a conv erging family of rig h t isosceles triangles inscrib ed into X with angles < π / 2. A similar argumen t also implies t hat on a torus T , the set U separates p oin ts ( y , z ) ∈ T with the corresp onding v ertex u inside of X , from those where u is outside. By con tinuit y , it suffices to sho w that the p oin t u crosses the edge of X as the generic p oin t ( y , z ) crosses U . Consider a p oint ( y , z ) ∈ U suc h that t he corresp onding t hird v ertex o f a square u lies in the relative interior of an edge e in X . No w fix y a nd c hange z . Since X is g eneric, p oint u will pass through the edge e , whic h implies the claim. W e need a few more observ ations on the structure of U . First, observ e that U do es not in tersect the diago nal ∆ = { ( y , y ) , y ∈ X } . Indeed, o therwise w e would hav e a sequence of inscrib ed right triangles ( y , z , u ) conv erging to the same p oin t , whic h is imp ossible since X do es not ha ve angles b et w een π / 2 and 3 π / 2 (see Figure 3 ). In a different direction, o bserv e that for a generic y the n umber o f p oin ts in U y is o dd. This follow s from t he previous a rgumen t and t he fa ct the num b er of in tersections of X and X ′ as ev en except at a finite num b er of p oin ts y . No w, from ab ov e w e can conclude that at least one o f the p o lygo ns in U is not null homotopic o n the torus T , since otherwise for a generic p oin t y t he size of U y is ev en. Fix one suc h p olygon and denote it by U ◦ . Since U ◦ is simple, not n ull homotopic and do es not inte rsect the diago nal ∆, w e conclude that U ◦ is homotopic to ∆ on Y . Therefore, there exist a con tinuous family of inscrib ed righ t isosceles triangles ( uy z ) suc h that when y g o es around X so do z and u . R elab eling triang les ( uy z ) with ( y z v ) w e o bt a in a simple p olygon V ◦ ⊂ V whic h is a lso homotopic to ∆ on T . Supp ose now that U ◦ and V ◦ do not in tersect. T ogether with ∆ these curv es sepa- rate the torus T in to three regions: region A betw een ∆ and U ◦ , region B b et w een U ◦ and V ◦ , and region C b et we en V ◦ and ∆ (see F igure 4). Consider the pairs ( y , z ) in the regions A a nd C whic h lie close t o ∆ (i.e. y and z lie close to eac h other on X ). Clearly , for suc h ( y , z ), either b oth corresponding p oin ts u and v lie inside X o r b oth u and v lie outside o f X . Let A b e the former and let C b e the latter regions. F rom a b ov e, for all ( y , z ) ∈ B we hav e u / ∈ X and v ∈ X . In other w ords, when y is fixed a nd z is mov ed along X coun terclo ck wise starting at y , o f the p oin ts u and v the first to mov e outside of X is alw a ys u . 4 IGOR P A K P S f r a g r e p la c e m e n t s A A B B C C V ◦ U U ◦ T T ∆ ∆ Figure 4. Set U on a tor us T a nd the sequence of regions A, B , C ⊂ T . No w, consider the smallest righ t equilateral tria ng le R inscribed in to X (the ex- istence was sho wn earlier). There are t w o wa ys to lab el it as shown in Figure 5. F or the first lab eling, if y is fixed and z is mo ved as ab ov e, the first t ime p oin t u lies on X is when z and u are v ertices of R . By a ssumptions on region A ⊂ T , the corresp onding p oint v lies inside X . Similarly , for the second lab eling, if y is fixed and z is mo ve d as ab ov e, t he first time p oint v lies on X is when z and v are v ertices of R . By assumptions on regions A ⊂ T , the corresp onding p oin t u lies outside of X , a contradiction. 1 P S f r a g r e p la c e m e n t s y y z z u u v v X X X R Figure 5. The smallest inscribed righ t isosceles triangle and its t wo lab elings. Finally , supp ose X is not generic. W e can p erturb the ve rtices of X to obtain a con tinuous fa mily of generic p olygons con verging to X and use the limit argumen t . Since X is simple, the conv erging squares do not disapp ear and conv erge t o a desired inscrib ed square. Similarly , when X has angles < π / 2 or > 3 π / 2, use the limit argumen t by cutting t he corners as sho wn in Figure 6. Figure 6. A conv erging family of obtuse p olygons. 1 The figure is somewhat misleading as it gives t he impressio n that for all y and z with | y z | s ma ller than that in R , we m ust hav e ( y , z ) ∈ A . In fact, we can hav e a ll these pair s in C and the same argument will work when A is substituted with C and the inside/outside pr oper ties ar e switched according ly . The p oin t is, by contin uity , a ll close ( y , z ) with a fixed order o n X determined by R , m ust lie in the sa me r egion (either A or C ). THE DISCRETE SQ UARE PEG PROBLEM 5 2. Proof by def orma tion In t his section, w e prov e the fo llowing extension of the main theorem: eve ry generic simple p olygon has an o dd num b er of inscrib ed squares. No w that we ha ve the relation, w e can try to prov e that it is in v ariant under certain elemen tary tr a nsitions. Theorem 2.1. Every generic simple p olygon has a n o dd numb er o f inscrib e d squar es. 2 The main theorem no w follows by a straightforw ar d limit argumen t. Note also that the theorem is false fo r al l simple po lygons; f or example ev ery right tr ia ngle has exactly t wo inscrib ed squares. W e b egin the pro of with the follo wing simple statemen t. Lemma 2.2. L et ℓ 1 , ℓ 2 , ℓ 3 and ℓ 4 b e four lines i n R 2 in gener al p o s ition. Th e n ther e exists a unique squar e A = [ a 1 a 2 a 3 a 4 ] such that x i ∈ ℓ i and A is oriente d clo ckwis e . Mor e over, the map ( ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 ) → ( a 1 , a 2 , a 3 , a 4 ) is c ontinuously differ entiable, wher e define d. 3 Pr o of. Fix z 1 ∈ ℓ 1 . Rotate ℓ 4 around z 1 b y π / 2, and denote b y ℓ ′ 4 the resulting line, and by z 2 = ℓ 2 ∩ ℓ ′ 4 the in tersection p oin t. Except when ℓ 2 ⊥ ℓ 4 , such z 2 is unique. Denote b y z 4 ∈ ℓ 4 the inv erse ro tation of z 2 around z 1 . W e obtain the right isosceles triangle ∆ = ( z 2 z 1 z 4 ) oriented clo ck wise in the plane. The fo ur t h v ertex z 3 of a square is uniquely determined. Start moving z 1 along ℓ 1 and observ e that the lo cus of z 3 is a line, whic h w e denote b y ℓ ′ 3 . Since line ℓ 3 is in general p osition with resp ect to ℓ ′ 3 , these tw o line inte rsect a t a unique p oint x 3 , i.e. determines uniquely the square [ a 1 a 2 a 3 a 4 ] as in the theorem. The second par t follo ws from immediately from the a b ov e construction.  Sketch of p r o of of The or em 2.1. W e b egin with the following restatemen t of the sec- ond part o f the lemma. Let X = [ x 1 . . . x n ] b e a generic simple p o lygo n and let { X t , t ∈ [0 , 1] } b e its con tinuous piecewise linear defor ma t ion. Supp ose A = [ a 1 a 2 a 3 a 4 ] is a n inscrib ed square with v ertices a i at different edges of X , and none at the vertice s of X , i.e. a i 6 = x j . Then, for sufficien tly small t , there exists a con tin uous defor ma t io n { A t } o f inscrib ed squares, i.e. squares A t inscrib ed in to X t . Moreo ve r, for sufficien tly small t , the vertice s a i of A t mo ve monotonically along the edges of X t . Consider what can happ en to inscrib ed squares A t as t increases. First, we ma y ha ve some non-generic p olygon X s , where suc h square in non- unique or undefined. Note that t he la tter case is imp ossible, since b y compactness we can alw ay s define a limiting square A s . If the piecewise linear deformation { X t } is c hosen generically , it is linear at time s , and we can extend the deformation o f A t b ey ond A s . The second obstacle is more delicate and o ccurs when the v ertex a i of square A s is at a ve rtex x j of X s . Clearly , w e can no longer deform A s b ey ond this p oin t. Denote b y e 1 the edge of X whic h contains v ertices a i of A t for t < s . Clearly , e 1 = ( x j − 1 , x j ) 2 It takes some effort to clar ify what we mean by a gener ic (see the pro of ). F o r now, the rea de r can read this as s a y ing that the n -gons, viewed as p o in ts in R 2 n , are almost s urely generic. 3 T o mak e this precise, think of lines ℓ i as points in RP 2 . 6 IGOR P A K or e 1 = ( x j , x j +1 ). D enote b y e ′ 1 the other edge adjacent to v . Denote by e 2 , e 3 and e 4 the other three edges of X con taining v ertices o f A t (see Figur e 7 ). P S f r a g r e p la c e m e n t s A t A s B r x j e 1 e ′ 1 e 2 e 3 e 4 Figure 7. Inscrib ed squares A t , A s = B s and B r , where t < s < r . Here e 2 , e 3 and e 4 are fixed, while e 1 and e ′ 1 mo ve aw ay fr o m the squares. No w consider a f a mily { B t } of squares inscrib ed in to lines spanned b y edges e ′ 1 , e 2 , e 3 and e 4 . By construction, A s = B s . There are t wo p ossibilities: either the corr esp ond- ing vertex b i approac hes x j from inside e ′ 1 or from the outside, when t → s and t < s . In the former case, w e conclude that the num b er of inscrib ed squares decreases by 2 as t passes through s . In the lat ter case, one square app ears a nd one disapp ears, so the parit y of the n um b er of squares remains the same. In summary , the parit y of the n umber of squares inscribed in t o X t with v ertices at differen t edges is in v arian t under the deformation. It remains to sho w that one can alw ays deform the p olygon X in suc h a w ay t hat at no p oin t in the deformation do there exist inscrib ed squares with more than one v ertex a t the same edge, and suc h that the resulting p olygon has an o dd n umber of inscrib ed squares. Fix a triangulatio n T o f X . Find a tria ngle ∆ in T with tw o edges the edges of X and one edge a diag onal in X . Sub divide the edges of X into small edges, so that neither of the new v ertices is a v ertex of an inscrib ed square. If the edge length is now small enough, w e can guar an tee that no square with tw o v ertices at the same edge is inscrib ed in to X . No w mov e the edges along tw o sides of the triangle ∆ tow ard the diagonal as sho wn in Figure 8. Re p eat the pro cedure. A t the end we o btain a p olygon Z with edges close to an interv a l. Observ e that Z has a unique inscribed square (see Figure 8). This finishes the pro of.  P S f r a g r e p la c e m e n t s X Z ∆ Figure 8. The first step of the p olygon deformation whic h preserv es the parit y of the num b er of inscrib ed squares; the final p olygon Z . THE DISCRETE SQ UARE PEG PROBLEM 7 3. The histor y, the proof ideas and the final remarks 3.1. The square p eg problem of inscribed squares has a long and in teresting history . It seems, every few y ears someone new falls in lo ve with it a nd works v ery hard to obtain a new v ariation o n the problem. Unfortuna t ely , as the results b ecome s tronger, the solutions b ecome more tec hnically in volv ed a nd sev eral of them start to include some gaps, still aw aiting careful scrutin y . 4 In terestingly , the impression one gets from the literature is that t ha t no direct elemen ta ry pro o f is ev en p ossible in t he piecewise linear case, as the problem is difficult indeed, the existing t echniq ues are inheren tly non-discrete and, presumably , other p eople hav e tried. 3.2. W e b egin with the celebrated incorrect pro of b y Ogilvy [F O ]. While the pro of w as refuted b y sev eral readers within a few months after its publication, it is still w orth going o ve r t his pro of and try find the gaps (there are three ma jo r ones, ev en if one assumes that the curv e is piecewise linear or analytic). As rep orted in [KW], Ogilvy later b ecame disillusioned in the p ossibilit y of a p ositiv e resolution of the problem. 3.3. The first ma j o r result w as pro ve d b y Emc h, who established the square p eg problem for con v ex curv es [E1]. La t er, Emc h writes in [E2] that T o eplitz and his studen ts disco vered the result indep enden tly tw o y ears earlier, in 191 1, but nev er published the pro of. W e refer to [G r ¨ u, p. 8 4] for further references to pro ofs in the con ve x case). Emc h starts by constructing a family o f inscrib ed rhom bi with a diagonal parallel to a g iv en line. By rotating the line and using uniqueness of suc h rhom bi he concludes that one can con tinuously rot ate a rhombus in to itself with tw o diagonals in terc hanged. Then the interme dia te v alue theorem implies that at some p oin t the rho mbus has equal diago na ls, thus giving a square. In t he larg ely forgott en follow up pa per [H], Hebb ert studies the squares inscrib ed in to quadrilaterals, essen tia lly pro ving Lemma 2.2. He stops short of applying his observ ations to general simple p olygons. Let us men tion also that in the second proof w e us e the fa ct that eve ry non-con v ex po lygon can b e triangula t ed. This is a standard result also due to Emc h [E2]. 3.4. An imp ortant breakthrough w as made b y Shnirelman in 1 9 29, when he o ffered a solution for curve s with piecewise con tinuous curv ature. This pap er w a s published in an obscure Russian publication, but later an expanded v ersion [Shn] was publis hed p osth umously . G uggenheimer in [G ug] studied this pro of, added and correct sev eral tec hnical points, and concluded that for Shnirelman’s pro of to w ork t he curv e nee ds to ha ve a b ounded v aria t io n. Shnirelman noted that for a generic curv e the parit y of the n umber of inscrib ed squares mu st b e inv arian t as the curve is deformed. The pro of uses a lo cal lemma on existence of inscrib ed square fo r closed curv es, a non-linear v ersion of Hebb ert’s observ ation (and, most like ly , completely indep enden t). F or the connectivit y of curv es with contin uous curv ature and b ounded v a riation Shnirelman 4 While we did find so me such rather unco n vincing ar gumen ts, we r efrain from commenting on them and leav e them to the exp erts. 8 IGOR P A K and G uggenheimer use kno wn adv anced results in the field. Fina lly , the fact that ev ery ellipse with unequal axis ha s a unique inscrib ed square is straightforw ard. Our pro of in Section 2 is mo deled on the deformation idea of Shnirelman (w e w ere una w are of Hebb ert’s pap er). In the piecewise linear case we no longer ha ve the analytic diffic ulties, but w e do get the unpleas an t obstacle of having inscrib ed squares with more than tw o v ertices on the same edge. In fact, if not for the smo oth case, there is no intuitiv e reason b ehind Theorem 2 .1 . In terestingly , w e b eliev e we kno w where Shnirelman got the idea of his pro of. A t the time of his first publicatio n, Shnirelman w as w orking with Lyusternik o n the conjecture of Poincar ´ e whic h states that ev ery smo oth con v ex surface has at least three closed geo desics. This conjecture w as made in the foundational pap er [P], whe r e P oincar´ e prov es that at least one suc h closed geo desics exists ( o n analytic surfaces), and this pro of uses a similar defo r ma t io n and pa rit y argumen t . 3.5. In 1961, Jerrard redisco vere d the square p eg problem and pr ov ed it for analytic curv es. He w as apparen tly motiv ated by the Kakutani’s theorem that ev ery conv ex b o dy has a circumscribed cub e. This result itself fo llo w ed a series of earlier similar results (see e.g. [Str]) and w as later extended b y Dyson, Floyd, and others. Jerrard’s pro of was a mo del of our pro of in Section 1. He similarly considers a curv e U on a torus T , corresp onding to inscrib ed rig ht isosceles triangles. He then uses a parit y a rgumen t to conclude that U is not n ull homot o pic, and a separate argumen t to conclude that when moving along U the fo ur t h v ertex cannot sta y on the same side of the p olygon. Our approa c h ha s sev eral adv antages due to the fact that w e can mak e them generic and thus a v o id squares whic h hav e to b e double coun ted. Also, w e use a straigh tforw ard ad ho c arg ument with the minimal inscrib ed triangle, different from that b y Jerrard. Ov erall, most details are still differen t due to the different nat ur e o f in t ersec tions of analytic and piecewis e linear curve s. 3.6. In recen t ye a rs, further results on the square p eg problem hav e app eared, most notably [St] and [Gri], whic h b oth w eak ened the restrictions on the curv es and ex- tended the reach of the theorem (to certain space quadrilaterals in [St] a nd to rect- angles in [Gri]). In fact, there is a long history of v ariatio ns o n the problem, whic h go es bac k to [Ka k]. Let us mention some of them. First, there ar e sev eral results on inscrib ed tria ngles and rhom bi and rectangles in general Jordan curv es [Ni1, NW]. W e refer to [Ni2] for the surv ey and further references. Second, there are sev eral results on cyclic quadrilaterals inscrib ed in to sufficien tly smo o th curv es [Ma1, Ma2, Ma3]. Note that in the piecewis e linear case, unless a quadrilateral Q ⊂ R 2 is an isosceles trap ezoid, one can alw a ys tak e a suffi- cien tly slim triangle X , suc h t ha t no p olygon similar to Q is inscribed into X . The corresp onding “isosceles trap ezoid p eg problem” is op en for general piecewise linear curv es. W e b eliev e that our pro of b y deformation might b e amenable to pro ve this result, but not without a ma jor c hange. In a differen t direction, an in teresting “ta ble theorem” in [F enn] say s that ev ery sufficien tly nice function f on a con v ex set U ⊂ R 2 has an a n inscrib ed square of giv en size, defined a s four p oin ts in U which are v ertices o f a square and hav e equal v alue THE DISCRETE SQ UARE PEG PROBLEM 9 of f . If t he graph of f is view ed as a t w o -dimensional hill, the insc rib ed square can b e in terpreted as feet of a square table, thus the name. Note that when the curv e C (in the square p eg problem) is a starred region, applying the ta ble theorem to the cone o ve r C giv es the desired square inscrib ed in to C . W e refer to [KK, Me1 , Me2, Me3] for mo r e on the ta ble theorem and other related results. Finally , there is a large n umber o f results extending the square p eg problem to higher dimensions, including curve s (see e.g. [W u]) and surfaces (see e.g. [HLM, Kra]). These results are to o num erous to b e listed here. W e refer to surv eys [CF G, Sec- tion B2] and [KW, Problem 11] for further references. 3.7. In conclusion, let us men tion tha t alt ho ugh stated differen tly , the results fo r man y classes of curv es are essen tially equiv alent. W e already saw this phenomenon in b oth pro ofs, where w e applied what w e called the limit ar gument . In each case, w e obtained one p olygon as the limit of others a nd noted that the sizes of inscrib ed squares do not con verge to zero. Of course, this approach fails in general, e.g. a rectifiable curv e can b e obtained as t he limit of piecewise linear curves , but a priori the inscrib es squares can collapse in to a p oin t. Nonetheless, one can use the limit argumen t in may cases that app ear in the lit- erature. It is easy to derive the square p eg problem for analytic curv es fr o m that of piecewise linear curv es. Similar ly , the piecewise linear curv es can b e obtained a a limit of analytic curv es a nd deriv e our main theorem f rom Jerrard’s pap er. It would b e in teresting to see how far the limit a r gumen ts take use from the piecewise linear curv es. Ac knowledgem en ts. W e are very gra t eful to Raman San yal for listening to the first sev eral ve rsions of these pro o fs, and to Ezra Miller for the encouragemen t. Sp e- cial thanks to Elizab eth D enne for the interesting discussions and sev eral helpful references. The author w as partially supp orted by the NSF grant. 10 IGOR P A K Reference s [CFG ] H. T. Croft, K. J. F a lconer and R. K. Guy , Unsolve d pr oblems in ge ometry , Springer , New Y o rk, 1991. [E1] A. Emch, So me prop erties of close d convex curves in a pla ne, Amer. J . Math. 35 (191 3), 407–4 12. [E2] A. Emch , On the Medians of a Closed Con vex P olygon, Amer. J . Math. 37 (1915), 19–28. [F e nn] R. F enn, The table theorem, Bul l. L ondon Math. 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