Fan distributions via Tverberg partitions and Gale duality

Equipartition theory, beginning with the classical ham sandwich theorem, seeks the fair division of finite point sets in $\mathbb{R}^d$ by the full-dimensional regions determined by a prescribed geometric dissection of $\mathbb{R}^d$. Here we examine $\textit{equidistributions}$ of finite point sets in $\mathbb{R}^d$ by prescribed $\textit{low dimensional}$ subsets. Our main result states that if $r\geq 3$ is a prime power, then for any $m$-coloring of a sufficiently small point set $X$ in $\mathbb{R}^d$, there exists an $r$-fan in $\mathbb{R}^d$ – that is, the union of $r$ ``half-flats’’ of codimension $r-2$ centered about a common $(r-1)$-codimensional affine subspace – which captures all the points of $X$ in such a way that each half-flat contains at most an $r$-th of the points from each color class. The number of points in $\mathbb{R}^d$ we require for this is essentially tight when $m\geq 2$. Additionally, we extend our equidistribution results to ‘‘piercing’’ distributions in a similar fashion to Dolnikov’s hyperplane transversal generalization of the ham sandwich theorem. By analogy with recent work of Frick et al., our results are obtained by applying Gale duality to linear cases of topological Tverberg-type theorems. Finally, we extend our distribution results to multiple $r$-fans after establishing a multiple intersection version of a topological Tverberg-type theorem due to Sarkaria.

💡 Research Summary

The paper introduces a new perspective on equipartition theory by shifting the focus from full‑dimensional regions to low‑dimensional subsets that “capture” a finite point set. The authors define an r‑fan as the union of r half‑flats (or half‑k‑flats) that share a common (k‑1)‑flat as a boundary; the codimension of the fan is d − k. In particular, a conical r‑fan has codimension r − 2 and is generated by r oriented hyperplanes whose normals are pairwise independent, with a non‑empty common intersection.

The main result (Theorem 1.1) states that for any prime‑power r ≥ 3 and any m‑coloring of a sufficiently small point set X⊂ℝ^{n−d−1}, where
n ≥ (r − 1)(d + m + 1) + 1,
there exists a conical r‑fan that equidistributes the coloring: each open half‑flat contains at most ⌈|X_i|/r⌉ points from each color class X_i. This bound is essentially tight for m ≥ 2 (Corollary 1.2).

A complex analogue (Theorem 1.3) replaces real space by ℂ^d and conical fans by complex regular r‑fans (regular r‑fans in ℝ^{2d} whose center is a complex affine hyperplane). The required number of points becomes (r − 1)(2d + m + 1) + 1, again essentially optimal.

The paper then treats piercing distributions in the spirit of Dolnikov’s theorem. Given a family ℱ of subsets of X, the condition χ(KG_r(ℱ)) ≤ m (the chromatic number of the r‑uniform Kneser hypergraph of ℱ) guarantees that ℱ can be partitioned into m subfamilies each avoiding r pairwise disjoint sets. Under this hypothesis, Theorem 1.4 asserts the existence of a conical (real case) or complex regular (complex case) r‑fan that not only distributes X but also pierces every A∈ℱ by at least two of its half‑flats.

The technical heart of the work is a Gale‑duality approach combined with topological Tverberg theorems. The authors first consider a linear map f : Δ_{n−1}→ℝ^d (or ℂ^d) and apply the topological Tverberg theorem for prime‑power r to obtain an r‑tuple of pairwise disjoint faces whose images intersect. By restricting to linear maps, this yields a combinatorial Tverberg partition of the vertex set. Applying the Gale transform to the configuration of points converts the Tverberg partition into a geometric fan in the original space. The “forbidden faces” condition (no set from ℱ is contained in any face of the partition) is enforced via the chromatic bound on KG_r(ℱ).

The authors further extend the method to multiple fans. Using a multiple‑intersection version of Sarkaria’s topological Tverberg theorem, they prove Theorem 1.7 and Theorem 1.8, which guarantee two r‑fans whose interiors intersect in regions each containing at most 1/r² of each color class. The proof reduces to a Borsuk‑Ulam‑type statement for the group ℤ_r ⊕ ℤ_r and is established via a Chern‑class calculation.

The paper is organized as follows: Section 2 reviews real and complex Gale transforms; Section 3 connects Tverberg partitions with fan distributions; Section 4 proves the single‑fan results; Sections 5–6 develop the two‑fan theorem (Theorem 1.8); Section 7 translates the two‑fan Tverberg result into the equidistribution statements of Theorem 1.7; and Section 8 discusses typical point configurations and sharpness of the bounds.

Overall, the work provides a unified framework that transforms topological Tverberg-type intersection results into concrete geometric partition statements involving low‑dimensional fans. It extends classical ham‑sandwich and Dolnikov theorems to a broad family of equipartition problems, covers both real and complex settings, and handles multiple fans, thereby significantly broadening the scope of equipartition theory.