Every generating polytope is strongly monotypic (Q6662765)

A convex polytope \(P\subset{\mathbb R}^n\) is called monotypic if every polytope with the same facet normals as \(P\) is combinatorially isomorphic to \(P\). It is called strongly monotypic if every polytope with the same facet normals generates by its facets an isomorphic arrangemant of hyperplanes. Both types of polytopes were introduced and investigated by \textit{P. McMullen} et al. [Geom. Dedicata 3, 99--129 (1974; Zbl 0283.52008)]. Among several other properties, they showed that a strongly monotypic polytope \(P\) is a generating set, that is, any nonempty intersection of two translates of \(P\) is a summand of \(P\). The converse was conjectured, but left open. The present paper now, finally, proves this conjecture. In view of an older result, it suffices to prove that every polytope that is monotypic but not strongly monotypic does not have the generating property. The proof is based on newly obtained characterizations of monotypic and strongly monotypic polytopes, for example: An \(n\)-dimensional convex polytope is strongly monotypic if and only if every \(n+1\) of its facet normals are not in conical position (conical position means that the vectors are separeted from \(0\) and none of them is in the positive hull of the others).