PreprintCC BY 4.0arXiv
A complete solution to the reverse isoperimetric problem for λ-convex bodies
Abstract
For a given λ> 0, a convex body in ℝn is called λ-convex if it is the intersection of balls of radius 1/λ. We show that among all λ-convex bodies in ℝn of a given surface area, volume is minimized by the λ-convex lens, that is, the intersection of two balls of radius 1/λ. This confirms a conjecture of Borisenko in all dimensions n ≥ 2. Moreover, we extend this reverse isoperimetric inequality to all model spaces of constant curvature in arbitrary dimension and arbitrary λ> 0.
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BibTeX
@misc{drach2026complete,
title = {{A complete solution to the reverse isoperimetric problem for $λ$-convex bodies}},
author = {Kostiantyn Drach and Kateryna Tatarko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40291},
eprint = {2610.40291},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40291},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A complete solution to the reverse isoperimetric problem for λ-convex bodies
AU - Kostiantyn Drach
AU - Kateryna Tatarko
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40291
UR - https://arxiv.org/abs/2610.40291
AB - For a given λ> 0, a convex body in ℝ^n is called λ-convex if it is the intersection of balls of radius 1/λ. We show that among all λ-convex bodies in ℝ^n of a given surface area, volume is minimized by the λ-convex lens, that is, the intersection of two balls of radius 1/λ. This confirms a conjecture of Borisenko in all dimensions n ≥ 2. Moreover, we extend this reverse isoperimetric inequality to all model spaces of constant curvature in arbitrary dimension and arbitrary λ> 0.
ER -