Three-dimensional polyhedra can be described by three polynomial inequalities

Abstract

Bosse et al. conjectured that for every natural number d ≥ 2 and every d-dimensional polytope P in ℝ d , there exist d polynomials p 1(x), pd (x) satisfying P = {x ∞ ℝd :p 1(x)≥ 0, pd (x) ≥ 0}. We show that every three-dimensional polyhedron can be described by three polynomial inequalities, which confirms the conjecture for the case d=3 but also provides an analogous statement for the case of unbounded polyhedra. The proof of our result is constructive. © 2009 Springer Science+Business Media, LLC.