Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909028893851648 |
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| author | Fryš, Filip |
| author_facet | Fryš, Filip |
| contents | This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the $C$-anisotropic $k$-dimensional Minkowski content of a $k$-rectifiable compact set always exists and coincides with a specific functional that depends naturally on $C$. We further show that the same conclusion holds for countably $\mathcal{H}^k$-rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the $C$-anisotropic $k$-dimensional Minkowski content for a countably $\mathcal{H}^k$-rectifiable compact set depends on the choice of $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets Fryš, Filip Classical Analysis and ODEs 28A75 This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the $C$-anisotropic $k$-dimensional Minkowski content of a $k$-rectifiable compact set always exists and coincides with a specific functional that depends naturally on $C$. We further show that the same conclusion holds for countably $\mathcal{H}^k$-rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the $C$-anisotropic $k$-dimensional Minkowski content for a countably $\mathcal{H}^k$-rectifiable compact set depends on the choice of $C$. |
| title | Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets |
| topic | Classical Analysis and ODEs 28A75 |
| url | https://arxiv.org/abs/2601.22681 |