Amalgamation of Regular Incidence‐Polytopes

We discuss the following problem from the theory of regular incidence‐polytopes. Given two regular d ‐incidence‐polytopes P 1 and P 2 such that the facets of P 2 and the vertex‐figures of P 1 are isomorphic to some regular ( d −1)‐incidence‐polytope K , is there a regular ( d +1)‐incidence‐polytope L with facets of type P 1 and vertex‐figures of type P 2 ?. Such amalgamations L of P 1 and P 2 along K exist (and then in fact very small ones) at least under the assumption that P 1 , and P 2 have the so‐called degenerate amalgamation property with respect to K . We prove some results on preassigning the ( d −1)‐dimensional medial section‐complex K for self‐dual ( d +1)‐dimensional regular incidence‐polytopes L .