PreprintCC BY 4.0arXiv
The isoperimetric problem for four-dimensional parallelohedra
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.
Related
- Constrained minimization problem and unbounded dynamics for the four-dimensional Klein-Gordon-Schrödinger System
- Double-arc rearrangement and the symmetric isoperimetric problem in Gauss space
- A comparison theorem for the planar strip isoperimetric problem
- The isoperimetric problem in the Riemannian Heisenberg $3$-space
- A complete solution to the reverse isoperimetric problem for λ-convex bodies
BibTeX
@misc{novaga2026isoperimetric,
title = {{The isoperimetric problem for four-dimensional parallelohedra}},
author = {Matteo Novaga and Lark Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01653},
eprint = {2610.01653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01653},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The isoperimetric problem for four-dimensional parallelohedra
AU - Matteo Novaga
AU - Lark Song
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.01653
UR - https://arxiv.org/abs/2610.01653
AB - 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.
ER -