On a Conjecture of A. J. Hoffman. Ii

It is proved that certain incidence relations of hyperplanes and closed convex sets in a rf-polytope can be preserved while replacing these sets by suitable polytopal subsets. The purpose of this paper is to prove Theorem 1. IfP is a d-polytope in Ed and Cx, * • •, Cj are closed convex subsets of P, such that every hyperplane that meets P meets (J¿=i Q> tnen there exist poly topes Dx, • • ■ , Dk with D¿c: C{ for all ll. Conjecture (3, 1, 3) is true (see Remark 4). Definitions. A polytope P is the convex hull of a finite set of points in the Euclidean o*-dimensional space Ed; a d-polytope in Ed is a polytope with nonempty interior; Vert F denotes here the set of vertices of a polytope P; if A<=E", conv A denotes the convex hull of A. An (affine) t-flat in Ed is a translate of a /-dimensional subspace of Ed, and a hyperplane is a (d— l)-flat. If His a hyperplane in Ed, then H+ and H_ (H+ and H_) denote the two closed (open, respectively) half-spaces of Ed, determined by H. Presented to the Society, January 23, 1971 ; received by the editors October 28, 1970. AMS 1969 subject classifications. Primary 5210, 5230; Secondary 5245.