Fan distributions via Tverberg partitions and Gale duality

TL;DR

This paper introduces new results on equidistributions of finite point sets in Euclidean space using $r$-fans, extending classical partition theorems through Gale duality and topological Tverberg-type theorems.

Contribution

It establishes the existence of $r$-fans that evenly distribute points of multiple colors in $ ext{R}^d$, extending classical fair division results to low-dimensional subsets.

Findings

Existence of $r$-fans capturing all points with balanced color distribution.

Extension of equidistribution results to piercing distributions.

Application of Gale duality and topological Tverberg theorems to distribution problems.

Abstract

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.