The Separation Problem, originally posed by K. Bezdek in , asks for the minimum number s(O, K) of hyperplanes needed to strictly separate an interior point O in a convex body K from all faces of K. It is conjectured that s(O, K) ≦ 2d in d-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensional polytopes.
Authors:N. Abrosimov, E. Makai, A. Mednykh, Yu. Nikonorov and G. Rote
We prove the theorem mentioned in the title for ℝn where n ≧ 3. The case of the simplex was known previously. Also the case n = 2 was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolic n-spaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas and some partial results about sufficient conditions for the existence of (convex) tetrahedra.