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Representing simple d-dimensional polytopes by d polynomials

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Abstract

A polynomial representation of a convex d-polytope P is a finite set {p 1(x), . . . , p n (x)} of polynomials over \({\mathbb {R}^d}\) such that \({P=\{x \in \mathbb {R}^d : p_i(x) \ge 0 \mbox{ for every }1 \le i \le n\}}\). Let s(d, P) be the least possible n as above. 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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Authors and Affiliations

  1. Faculty of Mathematics, University of Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Gennadiy Averkov & Martin Henk

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  1. Gennadiy Averkov
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  2. Martin Henk
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Correspondence to Gennadiy Averkov.

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The work of the authors was supported by the Research Unit 468 “Methods from Discrete Mathematics for the Synthesis and Control of Chemical Processes” funded by the German Research Foundation.

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Averkov, G., Henk, M. Representing simple d-dimensional polytopes by d polynomials. Math. Program. 126, 203–230 (2011). https://doi.org/10.1007/s10107-009-0280-y

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