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arXiv · 2109.03749

Lifting methods in mass partition problems

Pablo Soberón·Yuki Takahashi
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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.

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Pablo Soberón, Yuki Takahashi. 2021-09-08. Lifting methods in mass partition problems. https://arxiv.org/abs/2109.03749

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