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The isoperimetric problem for four-dimensional parallelohedra
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.01653
- Original
- Read it at the publisher
- Topics
Abstract
The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular $24$-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular $24$-cell also minimizes surface area among such tiles containing a fixed ball.
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