Directional curvature and medial axis
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915967447072768 |
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| author | Białożyt, Adam Bysiewicz, Dominik Denkowski, Maciej P. |
| author_facet | Białożyt, Adam Bysiewicz, Dominik Denkowski, Maciej P. |
| contents | The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C^1$-smooth hypersurfaces and extended to any codimension by Białożyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Directional curvature and medial axis Białożyt, Adam Bysiewicz, Dominik Denkowski, Maciej P. Metric Geometry The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C^1$-smooth hypersurfaces and extended to any codimension by Białożyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case. |
| title | Directional curvature and medial axis |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2604.26490 |