Cutting the same fraction of several measures

TL;DR

This paper investigates measure partition problems in Euclidean space, providing solutions for cutting equal fractions of multiple measures with hyperplanes and for finding convex subsets with prescribed measure values, under certain conditions.

Contribution

It introduces new results on measure partitioning and convex subset problems, extending previous work with specific assumptions for positive solutions.

Findings

Positive solutions for hyperplane cuts under additional assumptions

Existence of convex subsets with prescribed measure values

Extensions of measure partitioning techniques

Abstract

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.