Three-dimensional polyhedra can be described by three polynomial inequalities

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

2008-07-14

arxiv.org/abs/0807.2137v1

Mathematics

Metric Geometry

23 pages, 8 figures

Scientific paper

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.

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