On the (outer) Minkowski content with lower-dimensional structuring element
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917152779403264 |
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| author | Kiderlen, Markus Rataj, Jan |
| author_facet | Kiderlen, Markus Rataj, Jan |
| contents | Given a convex body $Q$ (structuring element) and a set $A$ in a Euclidean space, we consider the $Q$-Minkowski content of $A$. It is defined as the usual isotropic Minkowski content of $A$, but where the Euclidean ball is replaced by $Q$. When $Q$ is full-dimensional, the existence of the $Q$-Minkowski content can be assured by a sufficient condition which was stated by Ambrosio, Fusco and Pallara in the isotropic case. If $Q$ is not full-dimensional, we show that a weaker condition is sufficient for this purpose. We also consider the outer $Q$-Minkowski content of $A$ yielding the anisotropic perimeter of $A$ and we find a sufficient condition for its existence. Finally, we present an example of a set in three-dimensional Euclidean space, which does not admit the isotropic outer Minkwski content, but it admits the outer $Q$-Minkowski content for all two-dimensional disks $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the (outer) Minkowski content with lower-dimensional structuring element Kiderlen, Markus Rataj, Jan Metric Geometry 28A75, 49Q15, 52A39 Given a convex body $Q$ (structuring element) and a set $A$ in a Euclidean space, we consider the $Q$-Minkowski content of $A$. It is defined as the usual isotropic Minkowski content of $A$, but where the Euclidean ball is replaced by $Q$. When $Q$ is full-dimensional, the existence of the $Q$-Minkowski content can be assured by a sufficient condition which was stated by Ambrosio, Fusco and Pallara in the isotropic case. If $Q$ is not full-dimensional, we show that a weaker condition is sufficient for this purpose. We also consider the outer $Q$-Minkowski content of $A$ yielding the anisotropic perimeter of $A$ and we find a sufficient condition for its existence. Finally, we present an example of a set in three-dimensional Euclidean space, which does not admit the isotropic outer Minkwski content, but it admits the outer $Q$-Minkowski content for all two-dimensional disks $Q$. |
| title | On the (outer) Minkowski content with lower-dimensional structuring element |
| topic | Metric Geometry 28A75, 49Q15, 52A39 |
| url | https://arxiv.org/abs/2504.03339 |