PreprintarXiv licencearXiv
Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture
Abstract
We show that the fractional illumination number of every convex body in ℝd is at most 2d, with equality exactly for parallelotopes. We also prove that every such body can be covered by at most 2d(dlog d+dloglog d+O(d)) smaller positive homothetic copies as d→∞, establishing the optimal exponential rate in Hadwiger's covering conjecture. The proofs use a covering measure obtained by minimizing an overlap energy and a greedy covering argument on a finite net.
Related
BibTeX
@misc{gorodzha2026fractional,
title = {{Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture}},
author = {Yegor Gorodzha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23913},
eprint = {2609.23913},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23913},
}
RIS
TY - UNPB
TI - Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture
AU - Yegor Gorodzha
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23913
UR - https://arxiv.org/abs/2609.23913
AB - We show that the fractional illumination number of every convex body in ℝ^d is at most 2^d, with equality exactly for parallelotopes. We also prove that every such body can be covered by at most 2^d(dlog d+dloglog d+O(d)) smaller positive homothetic copies as d→∞, establishing the optimal exponential rate in Hadwiger's covering conjecture. The proofs use a covering measure obtained by minimizing an overlap energy and a greedy covering argument on a finite net.
ER -