Lifting methods in mass partition problems

TL;DR

This paper extends lifting methods in mass partition problems to polyhedral surfaces, proving new equipartition results for measures in Euclidean space using hyperplanes, spheres, and convex polytopes.

Contribution

It introduces a novel extension of lifting techniques to polyhedral surfaces, enabling new equipartition results for multiple measures in Euclidean spaces.

Findings

Existence of equipartitions of d+1 measures by parallel hyperplanes

Existence of equipartitions of d+2 measures by concentric spheres

Partitioning measures with convex polyhedral surfaces or polytopes with few facets or vertices

Abstract

Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of $d+1$ measures in $\mathbb{R}^d$ by parallel hyperplanes and of $d+2$ measures in $\mathbb{R}^d$ by concentric spheres. For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concentric spheres, convex polyhedral surfaces of few facets, or convex polytopes with few vertices.