2021/03/06 by Grigory Ivanov, Ivanov, Grigory, Márton Naszódi +1
Mathematics · #52A27 #52A35 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2103.04122
openalex publication_date 2021/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new variant of quantitative Helly-type theorems: the minimal "homothetic distance" of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the diameter. If K is the intersection of finitely many convex bodies in ℝd, then one can select 2d of these bodies whose intersection is of diameter at most (2d)3diam(K). The best previously known estimate, due to Brazitikos, is c d11/2. Moreover, we confirm that the multiplicative factor c d1/2 conjectured by Bárány, Katchalski and Pach cannot be improved.