TY - GEN A1 - Grötschel, Martin A1 - Henk, Martin T1 - On the Representation of Polyhedra by Polynomial Inequalities N2 - A beautiful result of Bröcker 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 $d(d+1)/2$. T3 - ZIB-Report - 02-15 KW - polyhedra and polytopes KW - semialgebraic sets KW - polyhedral combinatorics KW - polynomial inequalities Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6826 ER - TY - GEN A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial Inequalities Representing Polyhedra N2 - Our main result is that every $n$-dimensional polytope can be described by at most $2n-1$ polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an $n$-dimensional pointed polyhedral cone we prove the bound $2n-2$ and for arbitrary polyhedra we get a constructible representation by $2n$ polynomial inequalities. T3 - ZIB-Report - 04-53 KW - polyhedra and polytopes KW - semi-algebraic sets KW - polyhedral combinatorics KW - polynomial inequalities KW - stability index Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8284 ER - TY - GEN A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial Inequalities Representing Polyhedra N2 - Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound 2n-2 and for arbitrary polyhedra we get a constructible representation by 2n polynomial inequalities. T3 - ZIB-Report - 03-25 KW - polyhedra and polytopes KW - semi-algebraic sets KW - polyhedral combinatorics KW - polynomial inequalities KW - stability index Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7473 ER -