Fan distributions via Tverberg partitions and Gale duality
Shuai Huang, Jasper Miller, Daniel Rose-Levine, Steven Simon
TL;DR
The paper develops a comprehensive framework to equidistribute colored finite point sets in $\mathbb{R}^d$ using low-dimensional $r$-fans, extending classical equipartition results to conical (real) and complex regular (complex) fans for prime-power $r\ge3$. Central to the approach is Gale duality, which translates distribution questions into topological Tverberg-type problems; the authors leverage topological and cohomological tools, including Sarkaria-type theorems and a Borsuk-Ulam obstruction via Chern class computations, to obtain existence results for single and multiple fans. They establish real and complex versions of Dolnikov-type piercing theorems, characterize rainbow/distribution properties, and provide two-fan extensions with robustness and optimality insights for typical point configurations. The results yield new distribution phenomena (including rainbow and two-fan configurations) and connect geometric partition theory with deep topological constructs, offering sharp bounds and illustrating near-optimality in high dimensions. Overall, the work advances a unifying topological-combinatorial method for distributing colored point sets by narrow geometric objects and for understanding the limits of such distributions in both real and complex settings.
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.