Inscribed Tverberg‐type partitions for orbit polytopes. Issue 4 (5th September 2022) - Simon, Steven; Timofeyev, Tobias
Abstract: Tverberg's theorem states that any set of t ( r, d ) = ( r − 1 ) ( d + 1 ) + 1 t(r, d)=(r-1)(d+1)+1 points in R d R^d can be partitioned into r subsets whose convex hulls have non‐empty r ‐fold intersection. Moreover, generic collections of fewer points cannot be so divided. Extending earlier work of the first author, we show that one can nonetheless guarantee inscribed "polytopal partitions" with specified symmetry conditions in many such circumstances. Namely, for any faithful and full‐dimensional orthogonal representation ρ : G → O ( d ) ρ G→ O(d) of any order r group G, we show that a generic set of t ( r, d ) − d t(r, d)-d points in R d R^d can be partitioned into r subsets so that there are r points, one from each of the resulting convex hulls, which are the vertices of a convex d ‐polytope whose isometry group contains G via the regular action afforded by the representation. As with Tverberg's theorem, the number of points is optimal for this. At one extreme, this gives polytopal partitions for all regular r ‐gons in the plane, as well as for three of the six regular 4‐polytopes in R 4 R^4 . At the other extreme, one has polytopal partitions for d ‐polytopes on r vertices with isometry group equal to G whenever G is the isometry group of a vertex‐transitive d ‐polytope.