Equipartition of a measure by $(Z_p)^k$-invariant fans

TL;DR

This paper proves a topological result on partitioning measures in Euclidean space into equal parts using cones with a common vertex, specifically for dimensions that are odd prime powers, employing equivariant Euler class calculations.

Contribution

It introduces a new topological approach to measure partitioning in Euclidean spaces for dimensions that are odd prime powers, expanding the scope of equipartition results.

Findings

Partition of measures into 2d equal parts using cones is possible in certain dimensions.

The proof employs equivariant topology and Euler class calculations.

Results apply specifically when the dimension is an odd prime power.

Abstract

We prove a result about partitioning an absolute continuous measure in $\mathbb R^d$ into 2d equal parts by a system of cones with common vertex, where $d$ is an odd prime power. The proof is topological and based on the calculation of the equivariant Euler class of a certain vector bundle.