[Paper Review] Sierksma's Dutch Cheese Problem

K. S. Sarkaria|arXiv (Cornell University)|Mar 24, 1997

History and Theory of Mathematics6 references3 citations

TL;DR

This paper solves Sierksma's Dutch Cheese Problem by proving that the signed count of partitions of a generic set of $(q-1)(d+1)+1$ points in $\mathbb{R}^d$ into $q$ parts with intersecting convex hulls is always $((q-1)!)^d$. The result is derived using combinatorial topology and equivariant topology techniques, establishing a precise enumeration under sign-reversing symmetry.

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.

Motivation & Objective

• To resolve Sierksma's Dutch Cheese Problem, a long-standing open problem in discrete geometry concerning partitions of point sets in Euclidean space.
• To determine the signed count of partitions of a generic set of $(q-1)(d+1)+1$ points in $\mathbb{R}^d$ into $q$ parts whose convex hulls have a nonempty intersection.
• To establish a precise combinatorial invariant for such partitions using sign-reversing group actions and topological methods.
• To generalize known results on Tverberg-type theorems by introducing a signed enumeration that yields a fixed algebraic expression.

Proposed method

• Define a signed count of partitions of a generic point set in $\mathbb{R}^d$ into $q$ parts, where the sign is determined by the parity of the permutation induced by the partition.
• Use equivariant topology and the Borsuk–Ulam theorem to analyze the action of the symmetric group $S_q$ on the configuration space of partitions.
• Apply a topological degree argument to show that the total signed count is independent of the point configuration, provided the points are in general position.
• Leverage the fact that the configuration space deformation retracts to a complex with known homology, allowing computation of the signed sum.
• Use the structure of the order complex of the partition lattice to model the space of partitions and compute the Euler characteristic.
• Derive the final result by showing that the signed count matches the value $((q-1)!)^d$ via induction and symmetry reduction.

Experimental results

Research questions

• RQ1 What is the signed count of all partitions of a generic set of $(q-1)(d+1)+1$ points in $\mathbb{R}^d$ into $q$ parts whose convex hulls intersect?
• RQ2 Can the Tverberg-type partition problem be enumerated with signs such that the total sum is a fixed algebraic expression?
• RQ3 Does the signed count remain invariant under continuous perturbations of the point set, assuming general position?
• RQ4 Is there a topological invariant that captures the exact number of such partitions with sign, independent of the specific configuration?
• RQ5 Can the result $((q-1)!)^d$ be derived using equivariant cohomology or combinatorial topology techniques?

Key findings

• The signed count of all partitions of a generic set of $(q-1)(d+1)+1$ points in $\mathbb{R}^d$ into $q$ parts with intersecting convex hulls is exactly $((q-1)!)^d$.
• The result holds regardless of the specific configuration of the points, as long as they are in general position (generic).
• The sign-reversing action of the symmetric group $S_q$ on the set of such partitions leads to cancellation, leaving only the algebraic expression as the net count.
• The proof relies on topological invariance and the use of the Borsuk–Ulam theorem in the context of equivariant maps.
• The result generalizes classical Tverberg-type theorems by providing a signed enumeration rather than just existence.
• The value $((q-1)!)^d$ emerges as a topological invariant of the configuration space of partitions, reflecting the underlying symmetric structure.