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

TL;DR

This paper explores the concept of the degree of point configurations, linking it to Ehrhart theory, neighborly polytopes, and Tverberg points, providing classifications and structural insights into configurations of low degree.

Contribution

It offers a complete classification of degree 1 point configurations and a structural result for configurations with degree less than a third of the dimension, introducing weak Cayley decompositions.

Findings

Classified all degree 1 point configurations.

Established structure results for configurations with degree less than a third of the dimension.

Connected the degree concept to Tverberg points and the m-core of point sets.

Abstract

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 and 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 m-core of a set S of n points in R^r is contained in the set of Tverberg points of order 3m-2(n-r) of S.