On the representation of polyhedra by polynomial inequalities

Details On the representation of polyhedra by polynomial inequalities On the representation of polyhedra by polynomial inequalities

2002-03-26

arxiv.org/abs/math/0203268v2

Mathematics

Metric Geometry

19 pages, 4 figures; revised version with minor changes proposed by the referees

Scientific paper

A beautiful result of Br\"ocker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$.

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