2012/09/25 by Benjamin Nill, Nill, Benjamin, Arnau Padrol +1
Engineering · Mathematics · #52A35 #52B05 #52B11 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #graph theory and CDMA systems #math.CO #math.MG #msc:52A35 #msc:52B05 #msc:52B11
paper · pdf · doi:10.48550/arxiv.1209.5712
30 pages, 3 figures. Structure and presentation changes from v1
openalex publication_date 2012/09/25 · arxiv created 2013/08/27 · arxiv updated 2013/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 Rr is contained in the set of Tverberg points of order 3m-2(n-r) of S.