On Volume - Measure as Hemi - Metrics ∗

Abstract

Let m be a positive integer, and X be a point-set in a Euclidean space containing at least m + 2 points. Let μm : Xm+1 → [0,∞) be the map defined by μm(x1, . . . , xm+1) to be the m-dimensional volume of the convex hull of the point-set {x1, . . . , xm+1}. Denote by sm = sm(X) the maximum value of s such that s× μm(x1, . . . , xm+1) ≤ m+1 ∑ i=1 μm(x1, . . . , xi−1, xi+1, . . . , xm+2) holds whenever x1, x2, . . . , xm+2 are mutually distinct. If μm is identically zero, then put sm(X) = ∞. (This sm(X) is considered as a “bound” of the m-simplex inequality for μm on X.) We prove that if s2(X) = 3 and |X| ≥ 5, then X is the vertex set of a regular simplex, and present the values of sm of the vertex-sets of several convex polytopes. For the vertexset of the n-dimensional octahedron, we have sm = 3 for all n ≥ m ≥ 3, while for the vertex set of the n-dimensional cube, we show sm → 1 as n → ∞ for every m > 0. ∗Received November 30, 2004.

Cite this paper

@inproceedings{Deza2009OnV, title={On Volume - Measure as Hemi - Metrics ∗}, author={Michel Deza and MATHIEW DUTOUR and Hiroshi Maehara}, year={2009} }