2025/06/08 by Petar S. Kenderov, Kenderov, Petar, Олег Мушкаров +3
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC) #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2506.07252
openalex publication_date 2025/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ∂ C be the boundary of a compact convex body C in ℝn, n≥ 2, and O be an interior point of \mathcal C. Every straight line l containing O cuts from C a segment [AB] with end-points on ∂ C. It is shown that if [AB] is the shortest such segment, then ∂ C is smooth at the points A and B (i.e. at both of them there is only one supporting hyperplane for C) and, something more, the normals to the unique supporting hyperplanes at the points A and B intersect at a point belonging to the hiperplane through O which is orthogonal to [AB]. If C is a smooth compact convex body in ℝn, n≥ 2, the above property holds also when [AB] is the longest such segment. Similar results have place also when O is outside the set C. The ``local versions'' of these results (when the length |AB| of the segment [AB] is locally maximal or locally minimal) also have a place. More specific results are obtained in the particular case when C is a convex polytope.