Tverberg-Type Theorems with Trees and Cycles as (Nerve) Intersection Patterns

08/01/2018

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by   Jesús A. De Loera, et al.

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Tverberg's theorem says that a set with sufficiently many points in R^d can always be partitioned into m parts so that the (m-1)-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg's theorem is but a special case of a more general situation. Given sufficiently many points, all trees and cycles can also be induced by at least one partition of a point set.