Two properties of optimisers for the reverse isoperimetric problem
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912687765585920 |
|---|---|
| author | Hamdy, Deniz M. Scheuer, Julian |
| author_facet | Hamdy, Deniz M. Scheuer, Julian |
| contents | The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the boundary of the set is of class $C^{2}$, this amounts to a positive lower bound on the principal curvatures and in this class we prove that there are no $C^{2}$-maximisers of perimeter with prescribed volume. In addition, we prove that a given possibly non-$C^{2}$ maximiser has its smallest principal curvature constant in regions where it is of class $C^{2}$. We prove this result in the Euclidean, spherical and hyperbolic space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02688 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two properties of optimisers for the reverse isoperimetric problem Hamdy, Deniz M. Scheuer, Julian Differential Geometry Metric Geometry The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the boundary of the set is of class $C^{2}$, this amounts to a positive lower bound on the principal curvatures and in this class we prove that there are no $C^{2}$-maximisers of perimeter with prescribed volume. In addition, we prove that a given possibly non-$C^{2}$ maximiser has its smallest principal curvature constant in regions where it is of class $C^{2}$. We prove this result in the Euclidean, spherical and hyperbolic space. |
| title | Two properties of optimisers for the reverse isoperimetric problem |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2511.02688 |