On smoothness, tangent cones, and the metric geometry of definable sets
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911399602552832 |
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| author | Rocha, André Gadelha Sampaio, José Edson |
| author_facet | Rocha, André Gadelha Sampaio, José Edson |
| contents | In this paper, we present several definitive characterizations of the $C^1$ smoothness of definable sets in terms of their tangent cones and some other metric properties. In particular, we recover some of the beautiful characterizations presented by Ghomi and Howard (2014) and by Kurdyka, Le Gal, and Nhan (2018). For instance, we prove that for any $X\subset \mathbb{R}^n$ that is a locally closed $d$-dimensional definable set in an o-minimal structure, the following items are equivalent:
(1) $X$ is Lipschitz normally embedded (LNE), $C_3(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$ and depends continuously on $p$;
(2) For each $p\in X$, $X$ is Lipschitz regular at $p$ and $C_4(X,p)$ is a $d$-dimensional linear subspace;
(3) $X$ is a topological manifold and for each $p\in X$, $X$ is LNE at $p$ and $C_4(X,p)=C_3(X,p)$;
(4) $X$ is a topological manifold, and $C_5(X,p)$ is a $d-$dimensional subset for any $p\in X$;
(5) $X$ is $C^1$ smooth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On smoothness, tangent cones, and the metric geometry of definable sets Rocha, André Gadelha Sampaio, José Edson Metric Geometry Algebraic Geometry Differential Geometry Logic 14B05, 32S50 In this paper, we present several definitive characterizations of the $C^1$ smoothness of definable sets in terms of their tangent cones and some other metric properties. In particular, we recover some of the beautiful characterizations presented by Ghomi and Howard (2014) and by Kurdyka, Le Gal, and Nhan (2018). For instance, we prove that for any $X\subset \mathbb{R}^n$ that is a locally closed $d$-dimensional definable set in an o-minimal structure, the following items are equivalent: (1) $X$ is Lipschitz normally embedded (LNE), $C_3(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$ and depends continuously on $p$; (2) For each $p\in X$, $X$ is Lipschitz regular at $p$ and $C_4(X,p)$ is a $d$-dimensional linear subspace; (3) $X$ is a topological manifold and for each $p\in X$, $X$ is LNE at $p$ and $C_4(X,p)=C_3(X,p)$; (4) $X$ is a topological manifold, and $C_5(X,p)$ is a $d-$dimensional subset for any $p\in X$; (5) $X$ is $C^1$ smooth. |
| title | On smoothness, tangent cones, and the metric geometry of definable sets |
| topic | Metric Geometry Algebraic Geometry Differential Geometry Logic 14B05, 32S50 |
| url | https://arxiv.org/abs/2507.14950 |