2011/10/31 by Zsolt Lángi, Márton Naszódi, István Talata · 1 citation
Computer Science · Mathematics · #Convex analysis #Convex body #Convex combination #Convex hull #Convex polytope #Convex set #Convexity #Holomorphic and Operator Theory #Optimization and Variational Analysis #Orthogonal convex hull #Point processes and geometric inequalities #Subderivative #math.CO #math.MG #msc:52A30 #msc:52A35 #msc:52C17
paper · pdf · doi:10.1007/s00010-012-0160-z
27 pages, 5 figures
arxiv created 2012/09/05 · arxiv updated 2012/09/06 · openalex publication_date 2012/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let C⊂ \mathbb Rn be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ∅, or \mathbb Rn. The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm, that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.