The degree of point configurations: Ehrhart theory, Tverberg points and almost neighborly polytopes

The degree of a point configuration is defined as the maximal codimension of its interior faces. This concept is motivated from a corresponding Ehrhart-theoretic notion for lattice polytopes and is related to neighborly polytopes, to the Generalized Lower Bound Theorem and, by Gale duality, to Tverberg theory.The main results of this paper are a complete classification of point configurations of degree 1, as well as a structure result on point configurations whose degree is less than a third of the dimension. Statements and proofs involve the novel notion of a weak Cayley decomposition, and imply that the mm-core of a set SS of nn points in RrRr is contained in the set of Tverberg points of order (3m−2(n−r))(3m−2(n−r)) of SS.