Reverse Isoperimetric Properties of Thick $λ$-Concave Bodies in the Hyperbolic Plane
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911191395205120 |
|---|---|
| author | Esteban, Maria |
| author_facet | Esteban, Maria |
| contents | In this paper we address the reverse isoperimetric inequality for convex bodies with uniform curvature constraints in the hyperbolic plane $\mathbb{H}^2$. We prove that the\textit{ thick $λ$-sausage} body, that is, the convex domain bounded by two equal circular arcs of curvature $λ$ and two equal arcs of hypercircle of curvature $1 / λ$, is the unique minimizer of area among all bodies $K \subset \mathbb{H}^2$ with a given length and with curvature of $\partial K$ satisfying $1 / λ\leq κ\leq λ$ (in a weak sense). We call this class of bodies \textit{thick $λ$-concave} bodies, in analogy to the Euclidean case where a body is $λ$-concave if $0 \leq κ\leq λ$. The main difficulty in the hyperbolic setting is that the inner parallel bodies of a convex body are not necessarily convex. To overcome this difficulty, we introduce an extra assumption of thickness $κ\geq 1/λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reverse Isoperimetric Properties of Thick $λ$-Concave Bodies in the Hyperbolic Plane Esteban, Maria Metric Geometry Differential Geometry In this paper we address the reverse isoperimetric inequality for convex bodies with uniform curvature constraints in the hyperbolic plane $\mathbb{H}^2$. We prove that the\textit{ thick $λ$-sausage} body, that is, the convex domain bounded by two equal circular arcs of curvature $λ$ and two equal arcs of hypercircle of curvature $1 / λ$, is the unique minimizer of area among all bodies $K \subset \mathbb{H}^2$ with a given length and with curvature of $\partial K$ satisfying $1 / λ\leq κ\leq λ$ (in a weak sense). We call this class of bodies \textit{thick $λ$-concave} bodies, in analogy to the Euclidean case where a body is $λ$-concave if $0 \leq κ\leq λ$. The main difficulty in the hyperbolic setting is that the inner parallel bodies of a convex body are not necessarily convex. To overcome this difficulty, we introduce an extra assumption of thickness $κ\geq 1/λ$. |
| title | Reverse Isoperimetric Properties of Thick $λ$-Concave Bodies in the Hyperbolic Plane |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2411.09087 |