Bisecting measures with hyperplane arrangements

TL;DR

This paper proves that for a number of measures equal to a power of two in n-dimensional space, there exists an arrangement of hyperplanes that can bisect all measures simultaneously.

Contribution

It establishes a new bisecting theorem linking measure count and hyperplane arrangements in high-dimensional spaces.

Findings

Bisecting measures with hyperplane arrangements is possible when measures are a power of two.

The result applies to measures in any dimension, generalizing previous bisecting theorems.

Provides a constructive approach for hyperplane arrangements in measure bisecting.

Abstract

We show that when $n$ is a power of two, any $nD$ measures in $\mathbb{R}^n$ can be bisected by an arrangement of $D$ hyperplanes.