Lifting Methods in Mass Partition Problems

DOI10.1093/IMRN/RNAC224arXiv2109.03749OpenAlexW4292099387MaRDI QIDQ6182552

Publication date: 25 January 2024

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: Many results in mass partitions are proved by lifting

m a t h b b 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

m a t h b b R^{d}

by parallel hyperplanes and of

d + 2

measures in

m a t h b b 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.

Full work available at URL: https://arxiv.org/abs/2109.03749

zbMATH Keywords

mass partitions

Mathematics Subject Classification ID

Measure and integration (28-XX) Convex and discrete geometry (52-XX) Differential geometry (53-XX)