Typical Lipschitz images of rectifiable metric spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909367044931584 |
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| author | Bate, David Takáč, Jakub |
| author_facet | Bate, David Takáč, Jakub |
| contents | This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 $\mathcal{H}^n$-almost everywhere and, if $m>n$, preserves the Hausdorff measure of $E$. In general, we provide sufficient conditions, in terms of the tangent norms of $E$, for when a typical 1-Lipschitz map preserves the Hausdorff measure of $E$, up to some constant multiple. Almost optimal results for strongly $n$-rectifiable metric spaces are obtained.
On the other hand, for any norm $|\cdot|$ on $\mathbb{R}^m$, we show that, in the space of 1-Lipschitz functions from $([-1,1]^n,|\cdot|_\infty)$ to $(\mathbb{R}^m,|\cdot|)$, the $\mathcal{H}^n$-measure of a typical image is not bounded below by any $Δ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07943 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Typical Lipschitz images of rectifiable metric spaces Bate, David Takáč, Jakub Metric Geometry Functional Analysis 30L99 (Primary), 28A78 (Secondary) This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 $\mathcal{H}^n$-almost everywhere and, if $m>n$, preserves the Hausdorff measure of $E$. In general, we provide sufficient conditions, in terms of the tangent norms of $E$, for when a typical 1-Lipschitz map preserves the Hausdorff measure of $E$, up to some constant multiple. Almost optimal results for strongly $n$-rectifiable metric spaces are obtained. On the other hand, for any norm $|\cdot|$ on $\mathbb{R}^m$, we show that, in the space of 1-Lipschitz functions from $([-1,1]^n,|\cdot|_\infty)$ to $(\mathbb{R}^m,|\cdot|)$, the $\mathcal{H}^n$-measure of a typical image is not bounded below by any $Δ>0$. |
| title | Typical Lipschitz images of rectifiable metric spaces |
| topic | Metric Geometry Functional Analysis 30L99 (Primary), 28A78 (Secondary) |
| url | https://arxiv.org/abs/2306.07943 |