Algebraic Distance Optimization in Polyhedral Norms
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911611961212928 |
|---|---|
| author | Duarte, Eliana Kaihnsa, Nidhi Lindberg, Julia Torres, Angélica Weinstein, Madeleine |
| author_facet | Duarte, Eliana Kaihnsa, Nidhi Lindberg, Julia Torres, Angélica Weinstein, Madeleine |
| contents | We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the point and the inner normal fan of the polyhedral ball. For codimension-one varieties, we decompose $X$ into sets of points whose Voronoi cones have the same dimension, which is the expected dimension of their Voronoi cell. We prove that this decomposition is a stratification of $X$ and that each strata is a semialgebraic set. We conclude by giving an algebraic description of the medial axis, which is the locus of points whose minimal distance to $X$ is achieved at more than one point on $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Distance Optimization in Polyhedral Norms Duarte, Eliana Kaihnsa, Nidhi Lindberg, Julia Torres, Angélica Weinstein, Madeleine Algebraic Geometry Metric Geometry 51F99, 52Bxx, 49Q22, 14Pxx We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the point and the inner normal fan of the polyhedral ball. For codimension-one varieties, we decompose $X$ into sets of points whose Voronoi cones have the same dimension, which is the expected dimension of their Voronoi cell. We prove that this decomposition is a stratification of $X$ and that each strata is a semialgebraic set. We conclude by giving an algebraic description of the medial axis, which is the locus of points whose minimal distance to $X$ is achieved at more than one point on $X$. |
| title | Algebraic Distance Optimization in Polyhedral Norms |
| topic | Algebraic Geometry Metric Geometry 51F99, 52Bxx, 49Q22, 14Pxx |
| url | https://arxiv.org/abs/2604.19479 |