A reverse isoperimetric inequality in three-dimensional space forms
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| Format: | Preprint |
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2026
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| _version_ | 1866911498832445440 |
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| author | Drach, Kostiantyn Solanes, Gil Tatarko, Kateryna |
| author_facet | Drach, Kostiantyn Solanes, Gil Tatarko, Kateryna |
| contents | A $λ$-convex body in a three-dimensional space form $M^3(c)$ of constant curvature $c$ is a compact convex set $K$ whose boundary $\partial K$ has normal curvatures bounded below by a constant $λ>0$ (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all $λ$-convex bodies in $M^3(c)$, with a fixed surface area, the body of minimal volume is the $λ$-convex lens, i.e., the domain bounded by two totally umbilical caps of curvature $λ$. Moreover, this minimizer is unique. This result confirms Borisenko's Conjecture in the three-dimensional model spaces of constant curvature for $c\neq 0$, and complements recent progress on the conjecture in the Euclidean case $c=0$. As a by-product, our method also yields an alternative proof of the corresponding reverse isoperimetric inequality in two-dimensional hyperbolic space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_08132 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A reverse isoperimetric inequality in three-dimensional space forms Drach, Kostiantyn Solanes, Gil Tatarko, Kateryna Differential Geometry Metric Geometry A $λ$-convex body in a three-dimensional space form $M^3(c)$ of constant curvature $c$ is a compact convex set $K$ whose boundary $\partial K$ has normal curvatures bounded below by a constant $λ>0$ (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all $λ$-convex bodies in $M^3(c)$, with a fixed surface area, the body of minimal volume is the $λ$-convex lens, i.e., the domain bounded by two totally umbilical caps of curvature $λ$. Moreover, this minimizer is unique. This result confirms Borisenko's Conjecture in the three-dimensional model spaces of constant curvature for $c\neq 0$, and complements recent progress on the conjecture in the Euclidean case $c=0$. As a by-product, our method also yields an alternative proof of the corresponding reverse isoperimetric inequality in two-dimensional hyperbolic space. |
| title | A reverse isoperimetric inequality in three-dimensional space forms |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2603.08132 |