Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2301.07429 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913617255858176 |
|---|---|
| author | Rataj, Jan Winter, Steffen |
| author_facet | Rataj, Jan Winter, Steffen |
| contents | We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$ converge weakly to the surface area measure of the $r_0$-parallel set as $r\to r_0$. We further study the question which sets of parallel radii can occur as sets of non-differentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d=1$ and $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_07429 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On volume and surface area of parallel sets. II. Surface measures and (non-)differentiability of the volume Rataj, Jan Winter, Steffen Metric Geometry 28A75, 28A80, 51M25 We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$ converge weakly to the surface area measure of the $r_0$-parallel set as $r\to r_0$. We further study the question which sets of parallel radii can occur as sets of non-differentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d=1$ and $2$. |
| title | On volume and surface area of parallel sets. II. Surface measures and (non-)differentiability of the volume |
| topic | Metric Geometry 28A75, 28A80, 51M25 |
| url | https://arxiv.org/abs/2301.07429 |