Curvature extrema and four-vertex theorems for polygons and polyhedra

Curv ature extrema and four-v ertex theorems for p olygons and p oly he dra Oleg R. Musin Abstract Discrete a nalogs o f extrema of curv ature a nd generali zations of the four-vertex theorem to t he case of p olygo ns and p olyhedra are suggested and develo p ed. F or smo oth curves and p olygonal lines in the plane, a form ula relati ng the n umber of extrema of cu rv ature to the winding num- b ers of the curves (p olygonal lines) and their ev olutes is obtained. Also are considered higher-dimensional analogs of the four-vertex t heorem for regular and sh el lable t ria ngulations. 1 In tro duction 1.1. The notion of curv ature alwa ys play ed a sp e cial r ole in mathematics and theoretical physics. Despite the fact that this notion lives in science for mo re than tw o hundred years, the flow of pap ers devoted to it do es not draw, and the area of applications widens. In addition to the mathematica l asp ect, the sub ject of the present pap er has an entirely practical a spect. The distribution o f the cur v ature function a nd its s ingular p oin ts are of interest in many applied areas . The pr esen t autho r participated in pro jects related to cartogr aphic g eneralization (genera lization of the image of an ob ject on a map under passage to smaller scale), when it is necessary to distinguish singular po in ts on a curve or a surfac e: usua lly , they are zeros and (lo cal) extrema of curv ature [7, 13, 20]. Also, extr e ma of curv ature and their distribution ar e imp ortant characteristics in co mputer mo deling of surfaces. 1.2. In differential geo metry , the no tion of curv ature admits a “g oo d” definition only for the class o f ( C 2 -)smo oth curves and s ur faces. In the applications , one usually dea ls with discr e te mo dels, w her e a cur v e is repr esen ted by a p olygonal line and a s ur face is r e presen ted by a p olyhedron. Thus, there aris e s a problem of c orr e ct definition of the no tio ns of curv ature and extrema of curv ature in the discrete case. What do es the co rrectness mean in this case? T o our mind, the discr ete analog of curv ature must sa tisfy a cer tain g lobal prop erty of cur v ature fulfilled in the smo oth case. 1 1.3. O ne of such global pro p erties of the Ga uss curv ature is the Gauss–Bonnet theorem, the discr ete a na log of which is well known. Let M be a poly g on or a closed tw o-dimensiona l p olyhedron. W e a ssume that the cur v ature of M v an- ishes everywhere except the v ertices. In the first case, the ( angular ) curvatur e of M at a vertex A is the quantit y π − ∠ A, and in the seco nd case it is the quant ity 2 π − (the sum of the pla nar angles o f the faces of M at a vertex A ) . If M is a p olygon, then the sum of the curv atures at its vertices is equal to 2 π , and if M is a p olyhedron, then the sum is equal to 2 π χ ( M ), where χ ( M ) is the Euler character is tic of M . W e ea s ily se e that for such definition of curv ature we automatically obtain the same as sertion as in the Gauss– Bonnet theorem for the smo oth ca se. 1.4. T o o ur mind, another not les s importa nt pr operty of cur v ature is the four-vertex the or em for plane curves. Here it suffices to refer to V. I. Arnold, who said many times (and pr oved his words by numerous publications , see, e.g., [3, 4, 5, 6]), that this theorem reflects a fundamental pro perty of dimension tw o. W e star t o ur pa per with a short history o f the four- v ertex theorem (Section 2). “Discrete” versions of this theorem for p olygons are well known in g e ometry: it is famous lemmas b y Cauch y and A. D. Aleksandrov. 1.5. F or smo oth curves, the vertic es are critical p oin ts of the cur v ature func- tion on the curve. In the discrete ca se, where the curve is a clos ed poly gonal line, there are several definitions of the notion o f “vertex,” or, more precise ly , extremal vertex, see [15, 22, 23, 25]. In these pap ers, under the co r respo nding restrictions on the p olygon, discrete versions of the four-vertex theorem are ob- tained. In Sectio n 3, a definition of a n extr emal vertex of a p olygonal line is suggested such that the cor responding po in t of the ca ustic (evolute) is a cus p (a p oint o f r eturn). It turns out that ther e is a simple relationship betw een the num ber of extremal vertices and the winding n umbers of the curve and its caustic. A similar formula relates the num ber of the ta ngen t lines of a cur ve Γ drawn from a p oint P on the plane to the winding num ber of Γ and the deg ree of Γ with res p ect to P . 1.6. In Section 4, we consider versions o f the four -v ertex theo rem for regular and shellable triangula tions of a ball. Consider a d -dimensional simplicial p olytop e P in d -dimensiona l E uc lide a n space. W e call this p olytope ge neric if it has no d + 2 cos pherical vertices and is not a d -dimensiona l simplex. F rom now on, we co nsider only g eneric simplicia l po lytopes. Each ( d − 2)-dimensional face uniquely defines a neighboring sphere going thro ug h the vertices of tw o facets s haring this ( d − 2)-dimensiona l fa ce. Neighboring spher e is called empty if it does not contain other vertices of P and 2 it is called full if a ll other vertices of P are inside of it. W e ca ll an empty or full neighboring s phere extremal. Schatteman ([22], Theorem 2, p.232) claimed the following theorem: Theorem [Schat teman]. F or e ach c onvex d -dimensional p olytop e P ther e ar e at le ast d ( d − 2) -dimensional fac es defining empty n eighb oring spher es and at le ast d ( d − 2) -dimensional fac es defining ful l neighb oring s p her es. F or d = 2 this r esult is well known (see 2 .7). Although we do no t know whether this theo rem is fals e for d > 2, in [1] we show that Schatteman’s pro of ha s cer tain gaps. In the origina l version of this pap er we claimed that any regular triangulation of a conv ex d -p olytop e has at least d “ear s”. F or a pro of we us ed the same arguments as in [22]. Th us, the d - “ears” problem o f a regular triang ulation is still o pen. 2 F our-v ertex theorem and its generalizati ons 2.1. W e define a n oval a s a conv ex s mooth closed plane curve. The class ical four - vertex theorem by Muk hopadha ya ya [14] published in 1909 says the following: The curvatu r e function on an oval has at le ast four lo c al ex tr ema (vertic es). It is well known that any c o n tin uous function on a compact set has at leas t t wo (lo cal) extrema: a maximum and a minim um. It turns o ut that the curv a - ture function has at le a st four lo cal extrema. The pap er was noticed, a nd generaliza tions of the r esult app eared almost immediately . In 1912, A. Knes er show ed that con vexit y is not a necessar y condition and proved the four-vertex theorem for a simple closed plane cur v e. 2.2. The famous b o ok [9] b y W. Blaschke (first published in 1916 ), together with other generaliza tions, contains a “ relativ e” version of the four- vertex theorem. Here, we preser ve the fo rm ulation and notation from [9, p. 1 93]. L et C 1 and C 2 b e two (p ositively oriente d) c onvex close d curves, and let do 1 and do 2 b e ar c elements at p oints with p ar al lel (and c o dir e cte d) supp ort lines. Then t he r atio do 1 /do 2 has at le ast four extr ema. In the case where C 2 is a circle, this theorem turns into the theorem on four vertices of an ov al. 2.3. In 19 32, Bose [10] published a rema rk able version of the four-vertex theo- rem in the spirit of geometry “in the large.” While in the clas sical four-vertex theorem the extr e ma are defined “lo cally ,” here they are defined only “glo bally .” Let G be a n ov al no four p oin ts of which lie on a cir cle. W e denote by s − and s + (resp., t − and t + ) the num ber of its circles of curv a ture (resp., the cir cles touching G at exa ctly three po in ts) ly ing inside ( − ) and outside (+ ) the ov a l G , r espectively . (The cu rvatur e cir cle of G a t a point p touches G at p and has radius 1 /k G ( p ), where k G ( p ) is the curv ature of G at p .) In this notatio n, we hav e the relation 3 s − − t − = s + − t + = 2 . If w e define vertic es a s the p oin ts o f tangency o f the ov a l G with its circles of curv a ture lying entirely ins ide o r outside G , then these formulas imply that the ov al G has a t least four vertices. It is worth mentioning that this fact w as prov ed by H. Kneser for 10 years b efore B o se. (Actually , H. Knes er is a son of A. Kneser, and s o t wo interesting str engthenings of the four-vertex theor em belo ng to one family .) 2.4. Publica tions r e la ted to the four-vertex theor em did not stop from that time, and their num b er considera bly increased in the r e cen t years (see [3, 4, 5, 6, 19], etc.), to a large ex ten t owing to pap ers a nd talks by V. I. Arnold. In the ab o ve pap ers, v ario us versions of the four-vertex theorem for plane cur v es and convex curves in R d , and their s ing ular p oin ts (vertices) are co ns idered: cr itical p oin ts of the curv a ture function, flattening p oin ts, inflec tio n p oin ts, zero s of hig her deriv a tiv es, etc. In [2 4] there is a long list of pa pers devoted to this direction. 2.5. It is of interest that the first discr ete analog of the four-vertex theorem arose fo r almost 1 00 years b efore its smo oth version. (I use this opp ortunit y to thank V. A. Zalgaller , who brought this fact to m y attention.) In 1813, Cauch y , in his splendid pap er on rig idity of co n v ex p olyhedra, used the following lemma: Cauc h y l emma. L et M 1 and M 2 b e c onvex n -gons with sides and angles a i , α i and b i , β i , r esp e ctively. Assu me that a i = b i for al l i = 1 , . . . , n . Then either α i = β i , or the quantities α i − β i change sign for i = 1 , . . . , n at le ast four times. In Aleksandrov’s b o ok [2], the pr oof of the uniqueness of a co n v ex p olyhedron with given normals and areas of faces inv olves a lemma, where the angles in the Cauch y lemma a re replaced by the sides. W e present a version of it, which is somewhat less gener al than the or iginal one. A. D. Aleksandro v’s l emma. L et M 1 and M 2 b e two c onvex p olygons on the plane that have re sp e ctively p ar al lel sides. As s u me t hat no p ar al lel tr anslation puts one of them inside the other. Then when we p ass along M 1 (as wel l as along M 2 ), t he differ enc e of the lengths of the c orr esp onding p ar al lel sides changes the sign at le ast four times. W e easily se e the resemblance b etw een the ab o ve relative four-vertex theo rem for ov a ls (appar en tly b elonging to Blaschke) with the Ca uc h y and Aleksandr o v lemmas. F urthermor e, approximating ov als by po lygons, we easily prov e the Blaschk e theor e m with the help o f any of these lemma s . The Cauch y and Aleks andro v lemmas eas ily imply four-vertex theorems for a p o lygon. Corollary of the Cauc h y lemma. L et M b e an e quilater al c onvex p olygo n. Then at le ast two of the angles of M do n ot exc e e d the neighb oring angles, and at le ast two of the angles of M ar e n ot less than the neighb oring angles. Corollary of the Al eksandro v lemma. L et al l angles of a p olygo n M b e p air- wise e qual. Then at le ast two of the sides of M do n ot exc e e d their n eig hb oring 4 sides, and at le ast two of the sides of M ar e not less than t hei r neighb oring sides. 2.6. In the applications, the curvatur e r adi us at a vertex of a p olygon is usually calculated as follows. Consider a po lygon M with vertices A 1 , . . . , A n . Each vertex A i has tw o neig h bors : A i − 1 and A i +1 . W e define the cur v a ture r adius of M a t A i as follows: R i ( M ) = R ( A i − 1 A i A i +1 ) . Theorem [15]. Assume that M is a c onvex p olyg on and for e ach vertex A i of M t he p oint O ( A i − 1 A i A i +1 ) lies inside the angle ∠ A i − 1 A i A i +1 . Then the the or em on four lo c al extr ema holds t rue for the (cyclic) se quenc e of the nu m b ers R 1 ( M ) , R 2 ( M ) , . . . , R n ( M ) , i.e., at le ast two of the nu m b ers do not exc e e d the neighb oring ones, and at le ast two of the n umb ers ar e n ot less than the neighb oring ones. F urther mo re, this theorem generaliz es the four-vertex theor ems following from the Cauch y and Aleksandrov lemmas. 2.7. Discrete v ersion o f H. Kneser’s four-v ertex theorem. A cir c le C passing thro ugh certain vertices of a p olygon M is said to b e empty (resp ectively , ful l ) if all the remaining vertices o f M lie outside (resp ectiv ely , ins ide ) C . The circle C is extr emal if C is empty or full. Theorem [folklo re]. L et M = A 1 . . . A n b e a c onvex n -gon, n > 3 , no four vertic es of which lie on one cir cle. Then at le ast two of the n cir cles C i ( M ) := C ( A i − 1 A i A i +1 ) , i = 1 , . . . , n , ar e empty and at le ast two of them ar e ful l, i.e., ther e ar e at le ast four extr emal cir cles. (S. E . Ruks hin told the author that this r esult for many years is included in the list o f pro blems for training for mathematical comp etitions a nd is well known to St. Petersburg school students attending mathematical circles.) 2.8. It is a lso e asy to sugges t a direct gene r alization of the B o se theorem for the p olygons fro m the statement of Theorem 2.7. Theorem. We denote by s − and s + the n umb ers of empty and ful l cir cles among the cir cles C i ( M ) , and we denote by t − and t + the numb ers of empty and ful l cir cles p assing thr ough t hr e e p airwise nonn ei ghb oring vertic es of M , r esp e ctively. Then, as b efor e, we have s − − t − = s + − t + = 2 . The author sug gested this fa ct as a problem for the All-Rus s ia mathematics comp etition o f high-schoo l s tuden ts in 19 9 8. (The knowledge of Theorem 2.7 on four extremal circles, which is a corollar y of this fact, did not help muc h in solving this problem.) 2.9. One mo re generaliza tion of the Bos e theorem is given in [25], wher e one considers the case o f an e quilater al p olygon, which is not necess arily conv ex. 5 2.10. V. D. Sedy kh [23] prov ed a theor em on four supp ort plane s for strictly conv ex po lygonal lines in R 3 , and the ma in c o rollary of this theorem is also a version of Theorem 2.7 . Theorem [Se dyk h] If any two n eig hb oring vertic es of a p olygo n M lie on an empty cir cle, then at le ast four of the cir cles C i ( M ) ar e extr emal. It is clear that conv ex and equilater al po lygons sa tisfy this condition. F ur- thermore, Sedykh co nstructed exa mples o f p olygons showing that his theorem is wrong without this as sumption. 3 Extremal v ertices, cusps, and caustics 3.1. As already noted, the four-vertex theorem ho lds true for a n y smo oth s imple (i.e., without self-intersections) curve. Generally s peaking, w e canno t refuse from the ass umpt ion of simplicity o f the curve. F o r e x ample, the lemnisca te (figure eight) has only tw o vertices. W e obser v e that the winding num be r of the lemniscate v anishes. Ano ther ex ample: an ellipse has exac tly four vertices, and the winding num ber of the ellipse is equa l to 1. W e see that there is a certain r elationship be tw een the winding num be r of the curve and the num ber of vertices. One of the appro ac hes to the pro of of the four-vertex theor em for o v a ls inv olv es the relations hip be tw een vertices, cus ps, and the winding num ber of the ca ustic of a curve. In particular , this relatio nship is used in the pr oof in [15], as well as in a nu mber of as sertions in pap ers b y Arno ld (see [3 , 4] etc.). W e present a simple for m ula relating the num ber of vertices o f a plane curve with the winding num bers of the curve and its caustic, which also ho lds true for an arbitra ry close d p olygona l line. F or a p olygonal line, we define the notion of vertex (as p oin ts of lo cal extre mum of curv ature) without defining the cur v ature. A t the same time, this definition co mpletely corr e sponds to the no tion o f vertex in the smo oth case. 3.2. Let Γ b e a closed oriented p olygonal line in the plane w ith vertices V 1 , . . . , V n ordered cyclically . A vertex V i is said to b e p ositive if the left ang le at V (when we pass a long the p olygonal line Γ in accordance with the o rien tation) is at most π , Otherwise, V i is ne gative . Let C i = C ( V i − 1 V i V i +1 ). (F or s implicit y , we assume that no three sequential vertices of Γ lie o n one line.) Assume that a vertex V i is p ositive. W e say that the curvatur e at the vert ex V i is gr e ater (resp ectively , less ), than the curvatur e at the vertex V i +1 and write V i ≻ V i +1 (resp ectiv ely , V i ≺ V i +1 ) if the vertex V i +1 is p ositive a nd V i +2 lies outside (resp ectiv ely , inside) the circle C i , or the vertex V i +1 is negative, and V i +2 lies inside (resp ectively , outside) the circ le C i . In o rder to compar e the curv a tures at the vertices V i and V i +1 in the case where the vertex V i is negative, we repla ce in the ab o ve definitio n the word “grea ter ” by the word “ less,” and the word “outside” by the word “inside.” W e easily see that this definition is corre c t, i.e., that if V i ≻ V i +1 , then V i +1 ≺ V i . 6 3.3. Extremal vertices. A vertex V i of a poly gonal line is extr emal if the cur- v ature at V i is gr eater (resp ectively , less) than the curv a tures at tw o neighboring vertices, i.e., V i +1 ≺ V i ≻ V i − 1 or V i +1 ≻ V i ≺ V i − 1 . Remark 3.1. In [23], a vertex V i is said to b e su pp ort if al l vertic es of t he p olygonal line Γ lie t o one side fr om the cir cle C i . Ac c or ding to this definition, it would b e natur al to say that a vertex V i is an ext r emal (or a vertex of lo c al supp ort ) if b oth vertic es V i − 2 and V i +2 simultane ously lie inside or outs id e t he cir cle C i . However, such definition do es not always agr e e with t ha t pr esente d ab ove. It is e asy to give an ex amp le of a nonc onvex quadr angle having a vertex of lo c al supp ort which is not ext r emal. 3.4. Caustics. F ollowing Arnold, we define the c austic of a smo oth plane curve C as the set of centers o f cur v ature of the p oin ts of C . (In the class ical differential geometry , the ca ustic is called evolute .) By ana lo gy with the smo oth cas e, the caustic can b e defined for an arbitrar y closed p olygonal line Γ = V 1 . . . V n . W e s et O i = O ( V i − 1 V i V i +1 ) . Then the p olygonal line K (Γ) = O 1 . . . O n will b e called the c austic of the po lygonal line Γ. In the smo oth ca s e, the center of c ur v ature o f a v ertex o f a curve is a cusp of the caustic. Consider a p oint V of a smo oth c ur v e Γ with nonzero curv ature. Let V ′ , V ′′ ∈ Γ b e tw o p oin ts clo se to V lying on equa l distance h to different sides of V . W e observe that the po in t O ( V ′ V V ′′ ) tends to the center of c urv ature (i.e., to a p oin t of the ca ustic) as h → 0. W e easily see that | ∠ V ′ V V ′′ − ∠ O ( V ′ ) O ( V ) O ( V ′′ ) | = ( π if V is a vertex , 0 otherwise . Thu s, the a bsolute v alue of the difference b et ween the ang les at a vertex and at a cusp is equal to π . Accordingly , we define cusps o f the caustic K (Γ) for a p olygonal line. 3.5. Cusps. A vertex O i of the caustic K (Γ) is a cusp if ∠ V i − 1 V i V i +1 − ∠ O i − 1 O i O i +1 = ± π . Theorem 3.1. A vertex V i of the p olygo nal line Γ is extr emal if and only if the vertex O i is a cu sp of the c aust ic K (Γ) . This theo rem shows that the definition of e x tremalit y o f a vertex of a p olyg- onal line agree s with the c orresp onding definition in the smo o th case. 7 Pr o of. W e s imply consider all cas es of p ositiv eness and negativeness of the ver- tices V i − 1 and V i +1 and p osition o f the vertices V i − 2 and V i +2 with resp ect to the circle C i . 3.6. Winding num be r o f a curv e. Let Γ b e a closed oriented smo oth pla ne curve or a p olygonal line. (It is p ossible that Γ ha s self-intersections.) F or simplicity , we assume that Γ is generic, i.e., the smo oth cur v e has no rectilinear parts, and the p olygonal line has no three sequential vertices lie on o ne line. As b efore, an extremal vertex V of the curve Γ is said to b e p ositive , if the center of cur v ature o f V lies to the left with resp ect to the p ositive dir ection, and ne gative otherwise. W e denote b y N + (resp ectiv ely , N − ) the nu mber of po sitiv e (r espectively , of nega tiv e) extremal vertices of the curve Γ. F or Γ , the notio n of the winding num b er ind(Γ) is defined. F o r a p olygonal line, we can define the winding num ber as 1 2 π X i ( π − ∠ V i − 1 V i V i +1 ) , and for a n o rien ted smo oth curve – as the winding nu mber o f the tang en t vector of the curve, or as the quantit y 1 2 π Z k ( s ) ds, where k ( s ) is the curv ature a t the p oin t s . 3.7. The wi nding n um b er of the caustic of a curv e. In or der to formulate the main r esult of this section, we must define the notio n o f winding num ber als o for the caustic. In this case, the a bov e definition of the winding num b er formally do es not do b ecause for the po in ts o f the curve Γ at which the curv ature v anishes the center of c urv ature lies at “the p oint at the infinit y ,” and the tangent vector is not defined at cusps . How ev er, it is easy to see that in this case we can also c o rrectly define the winding num ber. W e only o bserv e that when we pass along the curve the rotation of the tangent vector of the caustic is contin uous everywhere (also including the p oin ts of the curve wher e the cur v ature v anishes), except the cus ps , where a jump by π o ccurs. The winding num b er of the c a ustic can b e rigo rously defined as the limit of the winding num ber of the caus tic of the p olygonal line uniformly approximating the c urv e. Theorem 3. 2. N + − N − = 2 ind(Γ) − 2 ind( K (Γ)) . Pr o of. First, we consider the case of a po lygonal line. If a vertex V i ∈ Γ is not extremal, then ∠ O i − 1 O i O i +1 = ∠ V i − 1 V i V i +1 . F or an extrema l vertex, thes e angles differ b y π (by − π ) if the vertex is p ositive (negative). Substituting these angles in the expressio n for the winding n umber of a po lygonal line, we obtain the r equired for m ula. 8 Approximating a s mo oth cur v e b y a p olygonal line, in the limit we obtain our formula and for a s mooth curve. How do es Theor e m 3 .2 r elate to the four -v ertex theorem? W e expla in this for the classical c a se of an ov al. W e observe that for an ov a l we hav e N − = 0 , and ind(Γ) = 1. Ther efore, the four-vertex theorem fo r an ov a l is obtained from Theorem 3.2 with the help o f the a dditional fact that ind( K (Γ)) < 0 (for the pro of see, e.g ., [15]). V. I. Arnold tur ned my attention to the fact that the ab ov e formula resembles the formula for the Maslov index of s pherical curves in his pap er [4]. It turns out that the list o f formulas o f such type can be contin ued. 3.8. Problem on the num ber of tangen t lines. Consider the problem on the n umber of the tange n t lines o f a cur v e pas s ing through a given point. Let Γ be a s mooth closed plane curve or a p olygonal line, and let p b e a po in t on the plane. W e consider the tangent lines dr a wn fro m p to Γ. If Γ is a p olygona l line, then a t angent line is a line l , passing thro ugh the vertex V o f Γ so that the vertices of Γ neighboring with V lie o n one side of l . As befor e, we give the sign of p oin ts of tang ency in a ccordance with the sign of the winding num ber of the curve, and we deno te by N + (b y N − ) the num b er of p ositiv e (negative) p oin ts of tangency . Here, ind p (Γ) deno tes the degr ee of the cur v e Γ with resp ect to p . Theorem 3. 3. N + − N − = 2 ind(Γ) − 2 ind p (Γ) . Pr o of. It is is similar to the pr oof of Theo rem 3.2. First, we prov e the as sertion for a p olygonal line, a nd then, by wa y of passag e to the limit, for a pla ne smo oth curve. T o our mind, the obvious simila r it y b et ween Theorems 3.2 and 3.3 and the Arnold for m ula shows that they are spec ia l cases o f a certain general fact on the num ber of tangent lines. 3.9. Case of p olyhedra. In co nc lus ion o f this section, we note tha t all definitions presented above for po lygonal lines are eas ily tra nsferred to the case of po lyhedra. Let M b e a simplicial d -dimensional p olyhedron immersed into R d +1 . The ( d − 1)-fa c es o f M play the role of the vertices V i , and the sphere containing a ll the d vertices of and the tw o vertices neighboring to the cor responding ( d − 1)- face plays the role of the circle C i . As in the cas e where d = 1, the sign of the angle of the face ( > or < π ) and p osition of the neighboring vertices (inside or outside) w ith r espect to these s pheres determine the order ≺ and ≻ for neig h- bo ring ( d − 1)-faces of M , and, therefor e, allow us to define (lo cal) ex tremalit y . In addition to extremals, for d > 1 there also may b e other “ singularities,” e.g ., “saddles,” for d = 2. It would b e interesting to find a g eneralization of Theorem 3.2 for the highe r -dimensional cas e . 9 4 Regular and shellable triangulations. Let D b e a simplicial d + 1 -polytop e, i.e. D is homeomor phic to a ( d + 1)-ball a nd all vertices of D lie on the b o undary . Let t b e a triangulation of D . Let us call an “ear” a bo undary s implex o f t . A precise definition of this pictur e s que term (whic h is a dopted in computationa l geo metry) is as follows: a ( d + 1 )-simplex s of t is a n “ e ar ” of t if at least tw o o f its d -faces lie on the bo undary o f D . How small can b e the n umber of “ea rs” of t ? First we co ns ider the simple ca se where d = 1. Prop osition 4.1. L et t b e a triangulation of an n -gon M , wher e n > 3 and al l vertic es of the triangulation ar e vertic es of M . Then the numb er of t he “e ars” of t he triangulation t exactly by 2 exc e e ds the numb er of the inner triangles. Pr o of. W e denote by x the num ber of “e a rs” in t , we denote by y the num ber of the tr iangles exa ctly one side of which is a side of M , a nd we denote by z the nu mber of the inner tr iangles. Then M has 2 x + y sides, i.e., 2 x + y = n. ( ∗ ) On the other hand, the total num b er of tria ng les is equal to n − 2, i.e., x + y + z = n − 2 . ( ∗∗ ) Subtracting ( ∗∗ ) from ( ∗ ), we obtain x − z = 2, as requir ed. Corollary 4.1 . Each t riangulation of a p olygo n M has at le ast two “e ars”. This a s sertion ca n b e regarded as a top ological version of the four -v ertex theorem, and the for m ula x − z = 2 can b e regar ded as a version of the Bose theorem. In the ge neral case, a tria ng ulation of the d -ball, wher e d > 2, may hav e no “ea rs.” In 1958 , M. E. Rudin [21] cons tr ucted the first example of such a triangulation for a conv ex po ly hedron in R 3 . Actually , she cons tr ucted a nonshel lable triangulation of the ball D 3 . (A triangulation t of the ball D d all vertices of which lie o n the bo undary of the ball is shel lable if the d -simplices s j in the tria ngulation t ca n b e ordered so that the union P k = s 1 ∪ s 2 ∪ . . . ∪ s k is homeomo rphic to the d -ball for a n y k = 1 , . . . , m . Here, m denotes the n um b er of d -simplices of t .) Prop osition 4.2. Any s hel lable triangulation of t he b al l D d has at le ast one “e ar”. Pr o of. Consider the simplex s m . Then, by definition, P m = P m − 1 ∪ s m , where P m and P m − 1 are homeomo rphic to the ball. This mea ns that s m is an “ e ar” bec ause other wise P m − 1 is not ho meomorphic to the ball. (Only the op eration of “cutting off a n ear” a llo ws us to deter mine shelling a triangula tion in the inv erse or de r .) 10 In par ticular, this pr opositio n implies that if a triangula tion ha s no “ea rs,” then it is not s he lla ble. W e complete this pape r by a result showing that the problem on the n umber of “ears” has a par tial solution in a rather general geo metric situation. Let S = { p i } ⊂ R d be a set consisting of n p oint s a nd let t be a tria ngulation of S . The triangula tion t is r e gular if there ex is ts a strictly conv ex piecewise- linear function, defined on the co n v ex hull of S a nd linear on ea c h simplex of t (see [8, 17]). W e e asily see that the Delaunay triang ulation of the s e t S is regular . Indeed, F or this pur pose, we set the v alues y i = k p i k 2 at vertices of the triangulation and contin ue the function to the s implices b y linearity . By the basic pr operty of the Delaunay tria ngulation [13], the function constructed is convex. It is known [12] tha t any regula r tria ng ulation is shella ble . Then Corollary 4. 2. Any r e gular tr iangulation of a generic c onvex simplicial p oly- he dr on has at le ast one “e ar”. The fact that regularity implies shellabilty als o shows that the triangula tion constructed b y Rudin is not regula r . F or other examples of irreg ular triangula- tions, see, e.g., [8]. Actually , using the metho d which is consider ed in [1 2, 22, 1] it can b e proved that any r e g ular triangula tio n t of a convex simplicial po ly tope ha s at least tw o “ears” . Moreover, t a nd its dual triangulation have together at least d + 1 ear s . A pro of o f this theorem will be considered in our further paper . 5 Conclusion One of the main aims of the present pap er was to s how that the progr ess in discrete versions of the four -v ertex theorem is not les s substantial than in the smo oth case. These theor ems hav e a lo ng history , which started with the Ca uc h y lemma. F ur thermore, in the discr ete cas e, there a re hig her-dimensional gener - alizations. Possibly , they w ill help to formulate a nd prove similar theo rems for the smo oth ca se. References [1] A. Akop y an, A. Glazyrin, O. R. Musin, and A. T a r asov, The E xtremal Spheres Theorem, ar Xiv:1003.3078 [2] A. D. Aleksa ndro v, Conv ex Polyhedra [in Russian], Moscow: GITTL, 1 9 50 [3] V. I. Arnold, T o pologic a l inv a rian ts of plane cur v es and ca ustics, Rutger s Univ. Lect. Serie s, V o l. 5 , Amer. Math. So c. P ro vidence, 1 994 [4] V. I. Arnold, Geometry of spherical cur ves and the quarternionic algebra, Usp. Mat. 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