Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

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.     

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.     

Diagonals of convex polytopes

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

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.     

The Mayer-Vietoris andIC equations for convex polytopes

In this paper we use geometric dissection to obtain linear equations on the flag vectors on convex polytopes. These results provide new proofs and expressions of the complete system of such equations originally discovered by Bayer and Billera. The Mayer-Vietoris equation applies to a situation where two convex polytopes overlap to produce union and intersection, both convex polytopes. The operatorsI andC applied to a polytope produce the cylinder (or prism) and cone (or pyramid), respectively, with the given polytopes as base. TheIC equation relates the flag vectors of the polytopes obtained in this way. As a consequence, it becomes easier to define linear functios of the flag vector, via initial data and their law of transformation under the operatorsI andC.     

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 Lagrange interpolation over lattices on a simplex

In this paper, a (d?+?1)-pencil lattice on a simplex in ${\mathbb{R}}^d$ is studied. The lattice points are explicitly given in barycentric coordinates. This enables the construction and the efficient evaluation of the Lagrange interpolating polynomial over a lattice on a simplex. Also, the barycentric representation, based on shape parameters, turns out to be appropriate for the lattice extension from a simplex to a simplicial partition.     

Covariance identities for normal variables via convex polytopes

Siegel (1993) presented a covariance identity involving normal variables that seems to flout notions of dependence. Here we show that it has an explanation from an unexpected quarter: convex geometry and the centroid known as the Steiner point.     

On convex envelopes for bivariate functions over polytopes

In this paper we discuss convex envelopes for bivariate functions, satisfying suitable assumptions, over polytopes. We first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two-dimensional polytope through the solution of a three-dimensional convex subproblem with continuously differentiable constraint functions. Then, for quadratic functions as well as for some polynomial and rational ones, again satisfying suitable assumptions, we show how the same computations can be carried out through the solution of a single semidefinite problem.     

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.     

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.     

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.