Representing simple d-dimensional polytopes by d polynomials

Details Representing simple d-dimensional polytopes by d polynomials Representing simple d-dimensional polytopes by d polynomials

2007-09-13

arxiv.org/abs/0709.2099v1

Mathematics

Metric Geometry

19 pages, 7 figures

Scientific paper

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of polynomials in a polynomial representation of P. It is known that d \le s(d,P) \le 2d-1. Moreover, it is conjectured that s(d,P)=d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P.

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