Three-dimensional polyhedra can be described by three polynomial inequalities

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.

💡 Research Summary

The paper addresses a conjecture made by Bosse, Grötschel, and Henk (2009) that every $d$‑dimensional polytope $P\subset\mathbb{R}^d$ can be described by exactly $d$ polynomial inequalities, i.e. there exist polynomials $p_0,\dots,p_{d-1}$ such that
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