Sierksma's Dutch Cheese Problem

TL;DR

This paper investigates a geometric partition problem in Euclidean space, establishing a signed count formula for partitions with intersecting convex hulls, and presents related results in combinatorial geometry.

Contribution

It provides a novel signed counting formula for partitions of Euclidean space subsets with intersecting convex hulls, extending previous combinatorial geometry results.

Findings

Signed count of partitions equals ((q-1)!)^d

Partitions with nonempty convex hull intersection are characterized

Related geometric and combinatorial results are presented

Abstract

Consider partitions, of a cardinality $(q-1)(d+1)+1$ generic subset of euclidean $d$-space, into $q$ parts whose convex hulls have a nonempty intersection. We show that if these partitions are counted with appropriate signs $\pm 1$ then the answer is always $((q-1)!)^d$. Also some other related results are given.