Flat regular polytopes

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP1 andP2, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP1 and whose vertex-figures are isomorphic toP2? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.