T. Bisztriczky

The University of Calgary | HBI · Department of Mathematics and Statistics

The Separation Problem asks for the minimum number s(O,K) of hyperplanes required to strictly separate any interior point O of a convex body K from all faces of K. The Conjecture is s(O,K) is at most 2 to the power d in real d-space , and we verify this for the class of simply linked neighbourly 4-polytopes.

The Separation Problem, originally posed by K. Bezdek in [1], asks for the minimum number s(O,K) of hyperplanes needed to strictly separate an interior point O in a convex body K from all faces of K. It is conjectured that s(O,K) 5 2d in d-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensio...

We prove the following: If a finite family of unit (radius) disks has the property that the distance between every pair of centres is greater than 4/3 and every subset of at most five disks has a common transversal line, then all disks have a common transversal line.

In this paper we show that every simple polygon can be triangulated with equal-diameter triangles. Our constructive proof does not give bounds for the number of triangles needed. We also show that every simple polygon can be partitioned into an infinite number of equal-perimeter triangles.

We classify the bicyclic polytopes and their vertex figures, up to combinatorial equivalence. These four-dimensional polytopes, which were previously studied by Smilansky, admit abelian groups of orientation-preserving symmetries that act transitively on their vertices. The bicyclic polytopes come in both simplicial and nonsimplicial varieties. It...

The separation number s(O, P) of a d-polytope with respect to a point O in the interior of P is the minimum number of hyperplanes necessary to strictly separate O from any facet of P. According to the Separation Conjecture, s(O, P) ≤ 2 d for d-dimensional convex polytopes in $${\mathbb{R}^d}$$ . We verify the Conjecture for totally-sewn 4-polytope...

An old question regarding the world we live in concerns what is real regarding points and lines: if two distinct lines intersect, is their intersection a unique point? In this paper, we take the approach that the answer is no, that all the points in the intersection are somehow close to one another (neighbourly) and that two non-neighbourly points...

A convex d-polytope in ℝ d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.

For a family $ \mathcal{F} $ \mathcal{F} of n mutually disjoint unit disks in the plane, we show that if any four disks are intersected by a line then there is a line that intersects at least n − 1 disks of $ \mathcal{F} $ \mathcal{F} .

Cyclic polytopes are characterized as simplicial polytopes satisfying Gale's evenness condition (a combinatorial condition on facets relative to a fixed ordering of the vertices). Periodically-cyclic polytopes are polytopes for which certain subpolytopes are cyclic. Bisztriczky discovered a class of periodically-cyclic polytopes that also satisfy G...

In a d-simplex every facet is a (d-1)-simplex. We consider as generalized simplices other combinatorial classes of polytopes, all of whose facets are in the class. Cubes and multiplexes are two such classes of generalized simplices. In this paper we study a new class, braxtopes, which arise as the faces of periodically-cyclic Gale polytopes. We giv...

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t k of the distances t for which any finite family of t-disjoint unit diam...

For a family F of n disjoint unit disks in the plane with the property T(4), we show that if there is a (n − 2)-transversal that strictly separates two elements of F then F has the property T − 1; that is, it has a (n − 1)-transversal. We apply this generic result to verify that T(4) implies T − 1 for families F of eight or nine disks.

A report on the Convex and Absract Polytopes Workshops held at the Banff International Research Station and The University of Calgary, May 19--22, 2005.

Let P denote a simplicial convex 2m-polytope with n vertices. Then the following are equivalent: (i) P is cyclic; (ii) P satisfies Gale’s Evenness Condition; (iii) Every subpolytope of P is cyclic; (iv) P has at least 2m+2 cyclic subpolytopes with n−1 vertices if n ≥ 2m+5; (v) P is neighbourly and has n universal edges. We present an additional ch...

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization o...

A convex 3-polytope in E 3 is edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope, and it is strictly antipodal if any two vertices lie on parallel planes that intersect the polytope in exactly one point. We prove that any edge-antipodal 3-polytope with six vertic...

Let P denote a finite set of points, in general position in the plane. In this note we study conditions which guarantee that P contains the vertex set of a convex polygon that has exactly k points of P in its interior.

We consider cyclic d-polytopes P that are realizable with vertices on the moment curve Md:t® (t,t2,¼,td)M_d:t\longrightarrow (t,t^2,\ldots,t^d) of order d ³ 3d\geq 3. A hyperplane H bisects a j-face of P if H meets its relative interior. For l ³ 1\ell\geq 1, we investigate the maximum number of vertices that P can have so that for some l\ell hyper...

For certain classes of neighbourly 4-polytopes P, any facet of P is strictly separated from an arbitrary fixed interior point of P by one of at most nine hyperplanes. This result, proved for the class of cyclic 4-polytopes by K. Bezdek and T. Bisztriczky, represents a verification of the Gohberg-Markus-Hadwiger Conjecture for the corresponding clas...

The centroid body. Recall that the support function of a compact convex set K is denned to be hK(u) = maxxΣk: {<u, x>}. The support function hK is positive homogeneous and convex, and any function with these properties is the support function of some compact convex set (see the illuminating paper of Berger [2], or the classic [5] by Bonnesen and Fe...

Based upon a labelling (total ordering) of vertices, a 4-polytope is Gale if the vertices satisfy a part of Gale's Evenness Condition and it is periodically-cyclic if there is an integer k such that every subset of k successive vertices generates a cyclic 4-polytope. Among the bi-cyclic 4-polytopes introduced by Smilansky, we determine which are Ga...

In 1982, I. Shemer introduced the sewing construction for neighbourly 2m-polytopes. We extend the sewing to simplicial neighbourly d-polytopes via a verification that is not dependent on the parity of the dimension. We present also descibable classes of 4-polyopes and 5-polytopes generated by the construction.

For each v, k and m such that v k 2m + 2 8, we construct a periodically-cyclic Gale 2m-polytope with v vertices and the period k. For such a polytope, there is a complete description of each of its facets based upon a labelling (total ordering) of the vertices so that every subset of k successive vertices generates a cyclic 2m-polytope.

A snake is a sequence of n congruent regular k-gons such that there are two end k-gons, each with one neighbour and n-2 remaining k-gons, each with two edge-to-edge neighbours. A snake is limited if it is not a proper subsnake of a bigger one. For k=3,4, and 6 the minimum values n of limited snakes are determined to be 20, 19, and 13, respectively.

A d -multiplex is a self-dual convex d -polytope whose quotient polytopes and faces are also multiplices. Presently, we verify that, up to 2d vertices, it has a unique underlying affine structure (oriented matroid).

A remarkable result of I. Shemer [4] states that the combinatorial structure of a neighbourly 2m-polytope determines the combinatorial structure of each of its subpolytopes. From this, it follows that every subpolytope of a cyclic 2m-polytope is cyclic. In this note, we present a direct proof of this consequence that also yields that certain subpol...

For eachk, m andn such thatn≥k≥2m+1≥5, we present a convex (2m+1)-polytope withn+1 vertices and 2( $2\left( {\mathop {k - m}\limits_m } \right) + \left( n \right)\left( {\mathop {k - m - n}\limits_{m - 1} } \right)$ ) facets with the property that there is a complete description of each of the facets based upon a total ordering of the vertices.

In 1957, H. Hadwiger conjectured that a convex body K in a Euclidean d-space, d ≥ 1, can always be convered by 2d smaller homotheti copies of K. We verify this conjecture when K is the polar of a cyclic d-polytope.

We recall that if S is a d - simplex then each facet and each vertex figure of S is a (d − 1)-simplex and S is a self-dual. We introduce a d-polytope P, called a d-multiplex, with the property that each facet and each vertex figure of P is a (d − 1)-multiplex and P is self-dual.

We present a comparative treatment of two approaches to homomorphisms between affine spaces of dimensionn≥2 that leads to characterizations of homomorphisms of these geometries that preserve dimension and parallelism.

Letn andd be integers,n>d 2. We examine the smallest integerg(n,d) such that any setS of at leastg(n,d) points, in general position in Ed, containsn points which are the vertices of an empty convexd-polytopeP, that is, SintP = 0. In particular we show thatg(d+k, d) = d+2k–1 for 1 k iLd/2rL+1.

In two previous papers we introduced the notion of an Affine Klingenberg space A and presented a geometric description of its free subspaces. Presently, we consider the operations of join, intersection and parallelism on the free subspaces of A: As in the case of ordinary affine spaces, we obtain the Parallel Postulate. The situation with join and...

We introduce a class of three-dimensional polytopesP with the property that there is a total ordering of the vertices ofP that determines completely the facial structure ofP. This class contains the cyclic 3-polytopes.

The Erds-Szekeres convex n-gon theorem states that for any integer n \ge 3, there is a smallest integer f(n) such that every set of at least f(n) points in the plane E, with no three on a line, contains the vertices of a convex n-gon. We consider three versions of this result as applied to convexly independent points and convex polytopes in E^d, d...

In a previous paper we introduced the notion of an affine Klingenberg space and described two equivalent axiom systems. The main tool for establishing this equivalence was the internal description of finite dimensional free subspaces. Our present axiom system yields the existence of maximal independent subsets and so we can introduce the notion of...

The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and fut...

LetK be a convex body in a Euclideand-spaceE d withd1. In 1957, H. Hadwiger conjectured thatK can always be covered by 2 d smaller homothetic copies ofK. We verify this conjecture in the case thatK is the polar of a cyclicd-polytope andd=3, 4 and 5.

We show that S E 2 contains a line segment illuminator if any two points of S are illuminated by a line segment of S in a given direction or if any eight points of S are illuminated by a connected set of line segments of S and a certain connectedness condition is fulfilled. We also show that if any three points of S E 2 are illuminated by a transla...

A limited snake of size n is a set of nonoverlapping unit disks D 1, ..., D nwith centers c 1, ..., c nwhere the distances ¦c ic j¦=2 if and only if ¦i−j¦=1, and no disk can touch D 1 or D nwithout further common points with D 1, ..., D nThe size of the smallest limited snake is proved to be 10.

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n=4 we show a fa...

A family of pairwise disjoint compact convex sets is called convexly independent, if none of its members is contained in the convex hull of the union of the other members of the family. The main result of the paper gives an upper bound for the maximum cardinalityh(k, n) of a family of mutually disjoint compact convex sets such that any subfamily of...

We call a convex subsetN of a convexd-polytopePE d ak-nucleus ofP ifN meets everyk-face ofP, where 0kd. We note thatP has disjointk-nuclei if and only if there exists a hyperplane inE d which bisects the (relative) interior of everyk-face ofP, and that this is possible only if [d+2/2]kd–1. Our main results are that any convexd-polytope with at mo...

It is proved that if ℱ is a family of nine pairwise disjoint compact convex sets in the plane such that no member of ℱ is contained in the convex hull of the union of two other sets of ℱ, then ℱ has a subfamily ℱ′ with five elements such that no member of ℱ′ is contained in the convex hull of the union of the other sets of ℱ′.

Without Abstract

Keywords: Helly-type theorems ; common transversals of plane compact convex sets Note: Professor Pach's number: [043] Reference DCG-ARTICLE-1988-002doi:10.1007/BF01313495 Record created on 2008-11-14, modified on 2016-08-08

A subset A of the real projective plane p2 S defined to be convex in p2, if there Is a line L in P~ disjoint from A and A is a convex subset of the anne plane P2\L; cf. [2] and [4]. Let d, be a finite collection of mutually disjoint convex sets in p2. In [3], N. H. Kuiper determines conditions under which there exists a line in p2 meeting every ele...

Let Γ be a differentiable curve in a real projective plane P 2 met by every line of P 2 at a finite number of points. The singular points of Γ are inflections, cusps (cusps of the first kind) and beaks (cusps of the second kind). Let n 1 (Γ), n 2 (Γ) and n 3 (Γ) be the number of these points in Γ respectively. Then Γ is non-singular if otherwise,...

: We prove a vertex theorem for space curves which need not lie on the boundaries of their convex hulls. 1. Introduction A regular closed simple curve in Euclidean 3-space, lying on the boundary of its convex hull and without zero curvature points, has at least four points where the torsion t vanishes. Under minor additional assumptions this was sh...

The classical four-vertex theorem states that a simple closed convex C ² curve in the Euclidean plane has at least four vertices (points of extreme curvature). This theorem has many generalizations with regard to both the curve and the topological space and for a history of the subject, we refer to [ 4 ] and [ 1 ]. The particular generalization of...

Let Φ be a regular closed C ² curve on a sphere S in Euclidean three-space. Let H ( S )[ H (Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H ( S ), let O ( p ) be the set of points of Φ whose osculating plane at each of these points passes through p. 1. THEOREM ([ 8 ]). If Φ has no multiple points and p ∈ H (Φ), then | 0 ( p ) | ≧ 3[4] when...

Let Γ be a differentiate curve in a real projective plane P2 intersected by every line at a finite number of points. A point of Γ is ordinary if Γ is locally convex at that point otherwise, the point is singular. Let the singular points of Γ consist of n1, inflections, n2 cusps (cusps of the first kind) and n3 beaks (cusps of the second kind). Then...

The problem of describing a surface of order three can be said to originate in the mid-nineteenth century when A. Cayley discovered that a non-ruled cubic (algebraic surface of order three) may contain up to twenty-seven lines. Besides a classification of cubics, not much progress was made on the problem until A. Marchaud introduced his theory of s...

How many of the continuous maps of a simple closed curve to itself are slope-preserving? For the unit circle S ¹ with centre (0, 0), a continuous map σ of S ¹ to S ¹ is slope-preserving if and only if σ is the identity map [σ(x, y) = ( x, y )] or σ is the antipodal map [ σ(x, y) = (– x , – y )]. Besides the identity map, more general simple closed...

A surface of order three F in the real projective three-space P ³ is met by every line, not in F , in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ 1 and τ 2 . In [ 2 ], we examined the biplanar surfaces containing the line τ 1 ⌒...

0. Introduction. A surface of order three, F , in the real projective threespace P ³ is met by every line, not in F , in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ 1 and τ 2 . In the present paper, we classify and describe tho...

A surface of order three F in the real projective three-space P ³ is met by every line, not in F , in at most three points. In the present paper, we determine the existence and examine the distribution of elliptic, parabolic and hyperbolic points; that is, the differentiable points of F which do not lie on any line contained in F .

A surfaceF of order three in the real projective three space is met by every line, not inF, in at most three points. We present a synthetic theory of these surfaces based upon a concept of differentiability. A pointv inF is a peak ifF has a unique tangent atvs and it does not lie inF. We classify theF with a peak by determining the number of lines...

A hypersurfaceS-1 of order two in the real projective n-space is met by every straight line in maximally two points; cf. [ 1, p. 391 ]. We develop a synthetic theory of these hypersurfaces inductively, basing it upon a concept of differentiability. We define the index and the degree of degeneracy ofS-1 and classify theS-1 in terms of these two quan...