2024/12/02 by Andreas F. Holmsen, Zuzana Patáková, Holmsen, Andreas F. +1
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces
paper · pdf · doi:10.48550/arxiv.2412.01445
A convex lattice set in ℤd is the intersection of a convex set in ℝd with the integer lattice ℤd. A classical theorem of Doignon states that the Helly number of d-dimensional convex lattice sets equals 2d, exponentially larger than the Helly number d+1 of ordinary convex sets in ℝd. By contrast, a remarkable theorem of Bárány and Matousek states that the fractional Helly number of convex lattice sets drops back down to d+1, matching the classical fractional Helly theorem of Katchalski and Liu. In this paper we generalize the Bárány--Matousek theorem to abstract convexity spaces (in the sense of van de Vel) that satisfy a suitable separation axiom. Our main result implies the following: if a separable convexity space has Radon number at most r, then its fractional Helly number is at most 2r. This bound is nearly tight, as illustrated by the case of box convexity in ℝd, whose Radon number is Θ(log d) and fractional Helly number equals d+1.