Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities
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), ..., p(d)(x) satisfying P = {x is an element of R(d) : p(1)( x) >= 0, ..., p(d)( 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.
polynomial polytope lojasiewicz's inequality semi-algebraic set theorem of brocker and scheiderer