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arXiv · 2504.03339

On the (outer) Minkowski content with lower-dimensional structuring element

Markus Kiderlen·Jan Rataj
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Abstract

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 $Q$.

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Markus Kiderlen, Jan Rataj. 2025-04-04. On the (outer) Minkowski content with lower-dimensional structuring element. https://arxiv.org/abs/2504.03339

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