Indecomposable convex polytopes

Shephard has given a criterion for the indecomposability (in the sense of Minkowski addition) of a convex polytope, in terms of strong chains of indecomposable faces joining pairs of vertices. Here, this criterion is weakened, to one involving strongly connected sets of indecomposable faces meeting every facet.     

Note dual polytopes of rational convex polytopes

LetP ^d be a rational convex polytope with dimP=d such that the origin of ^d is contained in the interiorP – P ofP. In this paper, from a viewpoint of enumeration of certain rational points inP (which originated in Ehrhart's work), a necessary and sufficient condition for the dual polytopeP ^dual ofP to be integral is presented.This research was performed while the author was staying at Massachusetts Institute of Technology during the 1988–89 academic year.     

Diagonals of convex polytopes

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 20–30, April, 1991.     

On empty convex polytopes

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 E^d, 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.     

Mixing regular convex polytopes

The mixing operation for abstract polytopes gives a natural way to construct a minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope, and completely determining the structure of the mix in each case.     

Nondegenerate families of convex cones and convex polytopes

This paper describes relations between convex polytopes and certain families of convex cones in R ⁿ .The purpose is to use known properties of convex cones in order to solve Helly type problems for convex sets in R ⁿ or for spherically convex sets in S _n , the n-dimensional unit sphere. These results are strongly related to Gale diagrams.     

Barycentric coordinates for convex polytopes

An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described. The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope. This polynomial, theadjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordinate functions for the polytope are rational combinations of adjoints of various dual cones associated with the polytope.     

Approximation of convex bodies by polytopes

LetC be a convex body ofE ^d and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n ^2/(d?1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.     

Addition and decomposition of convex polytopes

A new addition of convex polytopes is defined and the possibility of representing each polytope as a sum of “standard” polytopes is established The research reported in this paper was supported in part by the National Science Foundation NSF-G 19838, and by the Air Force Office of Scientific Research grant AF EOAR 63-63. Lecture delivered by the second author at a symposium on Series and Geometry in Linear Spaces, held at the Hebrew University of Jerusalem from March 16 till March 24, 1964.     

On a class of convex polytopes

Let ℘ denote the class of convex polytopesP having the following property: IfQ ₁ andQ ₂ are any subpolytopes ofP with no vertex in common, thenQ ₁ ∩Q ₂ is either empty or a single point. A characterization of ℘ is given which implies the characterization of strongly positively independent sets due to McKinney, Hansen and Klee.     

Classification of lattice-regular lattice convex polytopes

We completely describe lattice convex polytopes in ℝ ⁿ (for any dimension n) that are regular with respect to the group of affine transformations preserving the lattice. Supported in part by the RFBR (Grant Nos. SS-1972.2003.1 and 05-01-01012a) and the NWO-RFBR (Grant No. 047.011.2004.026/RFBR No. 05-02-89000-NWO_a).     

Approximation of convex sets by polytopes

The survey contains results related to different aspects of polyhedral approximation of convex bodies and some adjacent problems. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.     

Local tensor valuations on convex polytopes

Local versions of the Minkowski tensors of convex bodies in $n$ -dimensional Euclidean space are introduced. An extension of Hadwiger’s characterization theorem for the intrinsic volumes, due to Alesker, states that the continuous, isometry covariant valuations on the space of convex bodies with values in the vector space of symmetric $p$ -tensors are linear combinations of modified Minkowski tensors. We ask for a local analogue of this characterization, and we prove a classification result for local tensor valuations on polytopes, without a continuity assumption.