D-dimensional Convex Polytope Research Articles

Exact values and improved bounds on [formula omitted]-neighborly families of boxes

A finite family F of d-dimensional convex polytopes is called k-neighborly if d−k⩽dim(C∩C′)⩽d−1 for any two distinct members C,C′∈F. In 1997, Alon initiated the study of the general function n(k,d), which is defined to be the maximum size of k-neighborly families of standard boxes in Rd. Based on a weighted count of vectors in {0,1}d, we improve a recent upper bound on n(k,d) by Alon, Grytczuk, Kisielewicz, and Przesławski for any positive integers d and k with d⩾k+2. In particular, when d is sufficiently large and k⩾0.123d, our upper bound on n(k,d) improves the bound ∑i=1k2i−1di+1 shown by Huang and Sudakov exponentially.Furthermore, we determine that n(2,4)=9, n(3,5)=18, n(3,6)=27, n(4,6)=37, n(5,7)=74, and n(6,8)=150. The stability result of Kleitman’s isodiametric inequality plays an important role in the proofs.

A new proof of Balinski’s theorem on the connectivity of polytopes

Balinski (1961) proved that the graph of a d-dimensional convex polytope is d-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly d vertices; some of which appear to be new.

K-Sets and rectilinear crossings in complete uniform hypergraphs

In this paper, we study the d-dimensional rectilinear drawings of the complete d-uniform hypergraph K2dd. Anshu et al. (2017) [3] used Gale transform and Ham-Sandwich theorem to prove that there exist Ω(2d) crossing pairs of hyperedges in such a drawing of K2dd. We improve this lower bound by showing that there exist Ω(2dd) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K2dd. We also prove the following results.1.There are Ω(2dd3/2) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K2dd when its 2d vertices are either not in convex position in Rd or form the vertices of a d-dimensional convex polytope that is t-neighborly but not (t+1)-neighborly for some constant t≥1 independent of d.2.There are Ω(2dd5/2) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K2dd when its 2d vertices form the vertices of a d-dimensional convex polytope that is (⌊d/2⌋−t′)-neighborly for some constant t′≥0 independent of d.

Optimal recovery of three times differentiable functions on a convex polytope inscribed in a sphere

We consider the problem of global recovery on the class W3(P) of three times differentiable functions which have uniformly bounded third order derivatives in any direction on a d-dimensional convex polytope P inscribed in a sphere and containing its circumcenter. The information I(f) known about each function f∈W3(P) is given by its values and gradients at the vertices of P. The recovery error is measured in the uniform norm on P. We prove the optimality on the class W3(P) of a certain quasi-interpolating recovery method among all non-adaptive global recovery methods which use the information I(f). This method was constructed earlier for the case of a d-dimensional simplex T in the work by the author and T.S. Sorokina in 2011, where its optimality was proved for an analogous class W2(T) of twice differentiable functions.

Outer Normal Transforms of Convex Polytopes

For a d-dimensional convex polytope \(P\subset {\mathbb {R}}^d\), the outer normal unit vectors of its facets span another d-dimensional polytope, which is called the outer normal transform of P and denoted by \(P^*\). It seems that this transform, which clearly differs from the usual polarity transform, was not seriously investigated until now. We derive results about polytopes having natural properties with respect to this transform, like self-duality or equality for the composition of it. Among other things we prove that if P is a convex polytope inscribed in the unit sphere with the property that the circumradii of its facets are all equal, then \((P^*)^*\) coincides with P. We also prove that the converse is true when P or \(P^*\) has at most 2d vertices. For the three-dimensional case, we characterize the family of those tetrahedra T such that T and \(T^*\) are congruent.

The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes

We derive tight bounds for the maximum number of k-faces, $$0\le k\le d-1$$0≤k≤d-1, of the Minkowski sum, $$P_1+P_2$$P1+P2, of two d-dimensional convex polytopes $$P_1$$P1 and $$P_2$$P2, as a function of the number of vertices of the polytopes. For even dimensions $$d\ge 2$$d?2, the maximum values are attained when $$P_1$$P1 and $$P_2$$P2 are cyclic d-polytopes with disjoint vertex sets. For odd dimensions $$d\ge 3$$d?3, the maximum values are attained when $$P_1$$P1 and $$P_2$$P2 are $$\lfloor \frac{d}{2}\rfloor $$?d2?-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.

Convex hulls of spheres and convex hulls of disjoint convex polytopes

Given a set Σ of spheres in Ed, with d⩾3 and d odd, having a constant number of m distinct radii ρ1,ρ2,…,ρm, we show that the worst-case combinatorial complexity of the convex hull of Σ is Θ(∑1⩽i≠j⩽mninj⌊d2⌋), where ni is the number of spheres in Σ with radius ρi.To prove the lower bound, we construct a set of Θ(n1+n2) spheres in Ed, with d⩾3 odd, where ni spheres have radius ρi, i=1,2, and ρ2≠ρ1, such that their convex hull has combinatorial complexity Ω(n1n2⌊d2⌋+n2n1⌊d2⌋). Our construction is then generalized to the case where the spheres have m⩾3 distinct radii.For the upper bound, we reduce the sphere convex hull problem to the problem of computing the worst-case combinatorial complexity of the convex hull of a set of m disjoint d-dimensional convex polytopes in Ed+1, where d⩾3 odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the convex hull of a set of m disjoint d-dimensional convex polytopes in Ed+1 is O(∑1⩽i≠j⩽mninj⌊d2⌋), where ni is the number of vertices of the i-th polytope. Using the lower bound construction for the sphere convex hull problem, it is also shown to be tight for all odd d⩾3.Finally, we discuss how to compute convex hulls of spheres with a constant number of distinct radii, or convex hulls of a constant number of disjoint convex polytopes.

Separation in totally-sewn 4-polytopes with the decreasing universal edge property

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-polytopes with the property that the number of universal edges does not increase as the polytopes are constructed.

On the Graph Connectivity of Skeleta of Convex Polytopes

Given a d-dimensional convex polytope P and nonnegative integer k not exceeding d−1, let ${\mathcal{G}}_{k}(P)$ denote the simple graph on the node set of k-dimensional faces of P in which two such faces are adjacent if there exists a (k+1)-dimensional face of P which contains them both. The graph ${\mathcal{G}}_{k}(P)$ is isomorphic to the dual graph of the (d−k)-dimensional skeleton of the normal fan of P. For fixed values of k and d, the largest integer m such that ${\mathcal{G}}_{k}(P)$ is m-vertex-connected for all d-dimensional polytopes P is determined. This result generalizes Balinski’s theorem on the one-dimensional skeleton of a d-dimensional convex polytope.

On the f-vectors of the pyramidal triangulations of point configurations

A triangulation of a point configuration is called pyramidal if all its simplices have a vertex in common. A series of inequalities are derived for the components of the f-vectors of pyramidal triangulations, and, for every d ≥ 3, a d-dimensional convex polytope and its triangulation are constructed whose f-vector cannot be realized as the f-vector of a pyramidal triangulation.

Small convex polytopes with long edges and many vertices

We study how small the diameter of a d-dimensional convex polytope can be if its edges are not shorter than 1 and the number of its vertices is large. We give upper and lower bounds coinciding asymptotically in dimensions d = 3 and d ≥ 7 as the number of vertices tends to infinity.

Valuations on Polytopes Containing the Origin in Their Interiors

We give a classification of non-negative or Borel measurable, SL(d) invariant, homogeneous valuations on the space of d-dimensional convex polytopes containing the origin in their interiors. The only examples are volume, volume of the polar body, and the Euler characteristic.

Valuations on Polytopes Containing the Origin in Their Interiors

We give a classification of non-negative or Borel measurable, SL( d) invariant, homogeneous valuations on the space of d-dimensional convex polytopes containing the origin in their interiors. The only examples are volume, volume of the polar body, and the Euler characteristic.

Combinatorial face enumeration in convex polytopes

Let P be a d-dimensional convex polytope with n facets F 1, F 2,…, F n . The (combinatorial) representation of a face F of P is the set of facet indices j such that F ⊂ F j . Given the representations of all vertices of P, the combinatorial face enumeration problem is to enumerate all faces in terms of their representations. In this paper we propose two algorithms for the combinatorial face enumeration problem. The first algorithm enumerates all faces in time O( f 2 d min { m, n}), where f and m denotes the number of faces and vertices, respectively. For the case of simple polytopes, the second algorithm solves the problem in O( fd) time, provided that a good orientation of the graph of the polytope is also given as input.

Cellular strings on polytopes

The complex of cellular strings with respect to a generic linear functional on a d-dimensional convex polytope has the homotopy type of the ( d − 2 ) (d - 2) -sphere. This result was conjectured in a special case by H.-J. Baues.

The extended f-vectors of 4-polytopes

For P a d-dimensional convex polytope and S = { i 1,…, i s } ⊂ {0, 1,…, d−1}, let f s ( P) be the number of chains of faces Ø ⊂ F 1 ⊂ F 2 ⊂ … ⊂ F s ⊂ P with dim F j = i j . By the generalized Dehn-Sommerville equations the dimension of the affine span of the extended f-vectors ( f s ( P)) S ⊂ {0,1,2,3} as P ranges over all 4-polytopes is 4, and the extended f-vectors are determined by the values of f 0, f 1, f 2 and f 02. Six linear and four nonlinear inequalities on extended f-vectors of 4-polytopes are given. The consequences for the basic f-vector, ( f 0, f 1, f 2, f 3), are derived. These include the inequality, 4 f 2 ⩾ 3 f 0 − 10 + 7 f 3, conjectured by Barnette.

Another generalization of Carath�odory's theorem

Abstract : When P is a d-dimensional convex polytope with vertex-set V, the term V-simplex to denote a d-simplex whose vertices all belong to V is used. When k = 0 the theorem is Caratheodory's and when k = d-1 it is a slight sharpening of Goodearl's conjecture.

Polytopes, Valuations, and the Euler Relation

By a d-polytope we shall mean a d-dimensional convex polytope. We shall denote a j-dimensional face (or j-face) of a polytope by Fj. Every d-polytope P has proper j-faces for 0 ≦j ≦d — 1 and we shall also say that P is a d-face of itself. Observe that every face of a polytope is again a polytope. The collection of all convex polytopes shall be denoted by .

An Elementary Proof of Gram's Theorem for Convex Polytopes

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .

The number of faces of simplicial polytopes

The numbers of k-dimensional faces, f k≡f k(d), k=−1,0,…,d−1 , of a d-dimensional convex polytope satisfy the relations ∑ i − k d − 1 ( − 1 ) i f i ( i + 1 k + 1 ) = ( − 1 ) d − 1 f k with f −1=1, by convention. These relations are not independent and serve to determine (roughly) half of the f's in terms of the other half. Relations for the f's of upper index in terms of those of lower index found by Branko Grünbaum (in an unpublished report) are rederived here, and recurrences and interrelations of the coefficients in these relations are developed. In addition, the relations for the f's of even index in terms of those of odd index, and vice versa, are found