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Arranging convex bodies for maximum intersection volume: the sharp efficiency of centroid alignment — arXiv:2608.04516

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.04516
License (as archived): CC0-1.0
Archived: 2026-10-31T11:38:32.123Z
SHA-256 of the archived text: 9d8e025c3238dcd2a9a329df925c0b0dae28ca94ef04f8b4488c27dcfa01165a
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Arranging convex bodies for maximum intersection volume: the sharp efficiency of centroid alignment — arXiv:2608.04516 — arXiv (en), https://arxiv.org/abs/2608.04516 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_c970aeec6809 (Metraipolis archive dc_c970aeec6809, sha256 9d8e025c3238dcd2a9a329df925c0b0dae28ca94ef04f8b4488c27dcfa01165a)

BibTeX

@misc{dcc970aeec6809,
  title = {Arranging convex bodies for maximum intersection volume: the sharp efficiency of centroid alignment — arXiv:2608.04516},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.04516},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_c970aeec6809; sha256 9d8e025c3238dcd2a9a329df925c0b0dae28ca94ef04f8b4488c27dcfa01165a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.04516. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Given finitely many compact convex bodies in $\R^n$, one seeks translates maximizing the volume of their common intersection. A lazy solver merely translates each body so as to place its centroid at the origin. We prove that the lazy strategy always captures strictly more than $\left(\tfrac{2}{n+1}\right)^n$ of the optimal volume, and that this constant is sharp: families of cones over tangent disks, indexed by finite nets on the sphere, approach it. The infimum is not attained. For two convex bodies in the plane the resulting sharp constant $4/9$ closes a gap open since 1996, when de Berg, Cheong, Devillers, van Kreveld and Teillaud proved that centroid alignment of two convex polygons captures at least $9/25$ of the maximum overlap and exhibited examples capturing only $4/9$. The proof rests on the following identity: for a convex body $K$ with centroid at the origin, the intersection of all centroid-recentered compact convex supersets of $K$ equals $\tfrac{1}{n+1}(K-K)$. We close with a promise-problem variant in which the lazy strategy captures at least $\left(\tfrac{n}{n+1}\right)^n > \tfrac1e$ of the optimum, uniformly in the dimension. All results below have been checked in Lean~4; the one classical input quoted rather than proved is the equality case of the Brunn--Minkowski inequality, which enters only for $n\ge2$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-06 in math.OC).
Categorie arXiv: Metric Geometry (math.MG) ; Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2608.04516 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-31T11:28:34.335Z dalla pagina https://arxiv.org/catchup/math.OC/2026-08-06?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).