2015/11/12 by Silouanos Brazitikos, Brazitikos, Silouanos
Computer Science · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1511.07779
openalex publication_date 2015/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a new quantitative version of Helly's theorem: there exists an absolute constant α>1 with the following property: if \Pi: i∈ I\ is a finite family of convex bodies in \mathbb Rn with \rm int (\bigcapi∈ IPi )≠∅ , then there exist z∈ \mathbb Rn, s≤ αn and i1,… is∈ I such that z+Pi1∩⋯∩ Pis⊆ cn3/2(z+\bigcapi∈ IPi), where c>0 is an absolute constant. This directly gives a version of the "quantitative" diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound O(n3/2) can be improved to O(√(n)).