Dilation volumes of sets of finite perimeter
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866908761740804096 |
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| author | Kiderlen, Markus Rataj, Jan |
| author_facet | Kiderlen, Markus Rataj, Jan |
| contents | This paper analyzes the first order behavior (that is, the right sided derivative) of the volume of the dilation $A\oplus tQ$ as $t$ converges to zero. Here $A$ and $Q$ are subsets of $n$-dimensional Euclidean space, $A$ has finite perimeter and $Q$ is finite. If $Q$ consists of two points only, $x$ and $x+u$, say, this derivative coincides up to sign with the directional derivative of the covariogram of $A$ in direction $u$. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of $A$. We extend this result to finite sets $Q$ and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at zero. The proofs are based on approximation of the characteristic function of $A$ by smooth functions of bounded variation and showing corresponding formulas for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_09191 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Dilation volumes of sets of finite perimeter Kiderlen, Markus Rataj, Jan Probability 26B30, 28A75, 60D05 This paper analyzes the first order behavior (that is, the right sided derivative) of the volume of the dilation $A\oplus tQ$ as $t$ converges to zero. Here $A$ and $Q$ are subsets of $n$-dimensional Euclidean space, $A$ has finite perimeter and $Q$ is finite. If $Q$ consists of two points only, $x$ and $x+u$, say, this derivative coincides up to sign with the directional derivative of the covariogram of $A$ in direction $u$. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of $A$. We extend this result to finite sets $Q$ and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at zero. The proofs are based on approximation of the characteristic function of $A$ by smooth functions of bounded variation and showing corresponding formulas for them. |
| title | Dilation volumes of sets of finite perimeter |
| topic | Probability 26B30, 28A75, 60D05 |
| url | https://arxiv.org/abs/1708.09191 |