Three-dimensional polyhedra can be described by three polynomial inequalities

TL;DR

This paper proves that three-dimensional polyhedra can be exactly described by three polynomial inequalities, confirming a conjecture for dimensions up to three with a constructive proof.

Contribution

The authors demonstrate that for dimensions up to three, every polyhedron can be represented by exactly three polynomial inequalities, advancing the understanding of polynomial descriptions of polyhedra.

Findings

Three-dimensional polyhedra can be described by three polynomial inequalities.

The proof is constructive, providing explicit methods.

The result confirms a conjecture for dimensions up to three.

Abstract

Bosse et al. conjectured that for every natural number $d \ge 2$ and every $d$-dimensional polytope $P$ in $\real^d$ there exist $d$ polynomials $p_0(x),...,p_{d-1}(x)$ satisfying $P=\{x \in \mathbb{R}^d : p_0(x) \ge 0, >..., p_{d-1}(x) \ge 0 \}.$ We show that for dimensions $d \le 3$ even every $d$-dimensional polyhedron can be described by $d$ polynomial inequalities. The proof of our result is constructive.