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The Representation of Polyhedra by Polynomial Inequalities

Grötschel, Martin ; Henk, Martin

Springer

Discrete & computational geometry 29 (NaN), S. 485-504

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ISSN: 1432-0444

Quelle: Springer Online Journal Archives 1860-2000

Thema: Informatik , Mathematik

Notizen: Abstract. A beautiful result of Brocker 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 two- and three-dimensional case we get the expected number d(d+1)/2 .

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/s00454-003-0782-y

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