# Metric Thickenings, Borsuk-Ulam Theorems, and Orbitopes

## TL;DR

This paper explores the topology of metric thickenings of spheres, connecting geometric group actions, Borsuk-Ulam theorems, and zeros of trigonometric polynomials, providing new bounds and proofs.

## Contribution

It introduces new topological bounds for metric thickenings of spheres and generalizes Borsuk-Ulam theorems with optimal quantitative bounds.

## Key findings

- Homotopy type of metric thickenings of the circle determined.
- Weighted averages of odd maps on small sets are zero.
- Generalized Borsuk-Ulam theorems with optimal bounds.

## Abstract

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps $S^n \to \mathbb{R}^{n+2}$ on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps $S^{2n-1} \to \mathbb{R}^{2kn+2n-1}$. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.