Union of Translates of a Convex Body
Abstract
Let K⊂ℝd be a strictly convex body and let Z⊂ℝd be a finite set of sites in general position. We consider the problem of calculating the volume μd of ⋃z∈ Z(z−K). Let Vz be the Voronoi cell of z∈ Z with respect to the convex distance function induced by K, and let N be the nerve of the Voronoi cells. When K is a ball, a classical result reduces the inclusion-exclusion formula to a sum over N: μd(⋃z∈ Z(z−K))=∑J∈ N(−1)|J|+1μd(⋂z∈ J(z−K)). When may the hypothesis that K is a ball be dropped? We show that for d≤ 2 the reduced formula holds for every strictly convex K, while for each d≥ 3 there is a strictly convex K for which it fails. For x∈ℝd, let χx be the Euler characteristic of the subcomplex of N induced by the sites lying in K+x. A sufficient condition for the reduced formula to hold is χx≡ 1 on the union. When d=3 and K is C4, we show that χx≡ 1 on the union for all Z in general position if and only if K is an ellipsoid. The proof relies on the Fubini-Pick affine invariant.
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BibTeX
@misc{shelley2026union,
title = {{Union of Translates of a Convex Body}},
author = {Alec Shelley and Daniel Tartakovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25841},
eprint = {2610.25841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25841},
}
RIS
TY - UNPB
TI - Union of Translates of a Convex Body
AU - Alec Shelley
AU - Daniel Tartakovsky
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.25841
UR - https://arxiv.org/abs/2610.25841
AB - Let K⊂ℝ^d be a strictly convex body and let Z⊂ℝ^d be a finite set of sites in general position. We consider the problem of calculating the volume μ_d of ⋃_z∈ Z(z−K). Let V_z be the Voronoi cell of z∈ Z with respect to the convex distance function induced by K, and let N be the nerve of the Voronoi cells. When K is a ball, a classical result reduces the inclusion-exclusion formula to a sum over N: μ_d(⋃_z∈ Z(z−K))=∑_J∈ N(−1)^|J|+1μ_d(⋂_z∈ J(z−K)). When may the hypothesis that K is a ball be dropped? We show that for d≤ 2 the reduced formula holds for every strictly convex K, while for each d≥ 3 there is a strictly convex K for which it fails. For x∈ℝ^d, let χ_x be the Euler characteristic of the subcomplex of N induced by the sites lying in K+x. A sufficient condition for the reduced formula to hold is χ_x≡ 1 on the union. When d=3 and K is C^4, we show that χ_x≡ 1 on the union for all Z in general position if and only if K is an ellipsoid. The proof relies on the Fubini-Pick affine invariant.
ER -