Lifting methods in mass partition problems
Abstract
Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higherdimensional space and dividing the higherdimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.03749
 Bibcode:
 2021arXiv210903749S
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 16 pages, 3 figures