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 ${\mathrm{\Re }}^d$ there exist $d$ polynomials $p_0(x),\mathrm{.}\mathrm{.}\mathrm{.},p_{d-1}(x)$ satisfying $P={x \in $\mathbb{R}$^d : p_0(x) \ge 0,

…, p_{d-1}(x) \ge 0 }.$Weshowthatfordimensions$d \le 3$evenevery$d$-dimensionalpolyhedroncanbedescribedby$d$ polynomial inequalities. The proof of our result is constructive.