Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

TL;DR

This paper extends the Borsuk-Ulam theorem to Stiefel manifolds and applies it to derive bounds on the dimension for orthogonal hyperplanes that partition measures equally.

Contribution

It generalizes the Borsuk-Ulam theorem to Stiefel manifolds and establishes new bounds for orthogonal mass partitions in Euclidean spaces.

Findings

Derived bounds on dimension d for orthogonal hyperplanes partitioning measures

Extended Borsuk-Ulam theorem to Stiefel manifolds

Established stronger conditions for measure partitioning with orthogonal hyperplanes

Abstract

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal hyperplanes, any $n$ of which partition each of the measures into $2^n$ equal parts. If $n=k$, the result corresponds to the bound obtained in [11], but with the stronger conclusion that the hyperplanes are mutually orthogonal.