Locally projective regular polytopes
The 2-sections of a finite regular polytope are polygons, i.e., they are spherical. If all the sections are spheres, then the combinatorially regular polytopes are obtained, which are known to be isomorphic to the convex regular polytopes. A natural question is the following: what happens if the minimal sections are of some other (sufficiently simple) topological type? If these minimal sections are all that is needed to determine the polytope, then the polytope is universal of its type. So a regular polytope is said to be locally projective if its minimal sections which are not spherical are projective spaces, and if these sections which are not spherical are projective spaces, and if these sections (together with the Schläfli type) determine the polytope. For each \(n\geq 4\), a family of \(n\) or \(n-1\) (as \(n\) is odd or even) locally projective regular polytopes is described, and these polytopes are shown to be universal of their type.
scientific article; zbMATH DE number 529903