Measure Partitions Using Hyperplanes with Fixed Directions

TL;DR

This paper investigates measure partitions in high-dimensional spaces using hyperplanes with fixed directions, generalizing classical results and providing new optimal mass-partitioning methods.

Contribution

It introduces new bounds and methods for measure partitions with fixed hyperplane directions, extending classical and high-dimensional partitioning results.

Findings

Number of measures that can be evenly split using fixed-direction hyperplanes

Existence of paths with limited turns that halve multiple measures in the plane

Optimal mass-partitioning strategies for chessboard colorings in high dimensions

Abstract

We study nested partitions of $R^d$ obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among $k$ sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any $t$ measures there is a path formed only by horizontal and vertical segments using at most $t-1$ turns that splits them by half simultaneously, and optimal mass-partitioning results for chessboard-colourings of $R^d$ using hyperplanes with fixed directions.