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arXiv · 2604.19479

Algebraic Distance Optimization in Polyhedral Norms

Eliana Duarte·Nidhi Kaihnsa·Julia Lindberg·Angélica Torres·Madeleine Weinstein
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Abstract

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 $X$.

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Eliana Duarte, Nidhi Kaihnsa, Julia Lindberg, Angélica Torres, Madeleine Weinstein. 2026-04-21. Algebraic Distance Optimization in Polyhedral Norms. https://arxiv.org/abs/2604.19479

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