The fractional Helly number for separable convexity spaces

Abstract A convex lattice set in $$\mathbb {Z}^d$$ Z d is the intersection of a convex set in $$\mathbb {R}^d$$ R d with the integer lattice $$\mathbb {Z}^d$$ Z d . A classical theorem of Doignon states that the Helly number of d -dimensional convex lattice sets equals $$2^d$$ 2 d , exponentially larger than the Helly number $$d+1$$ d + 1 of ordinary convex sets in $$\mathbb {R}^d$$ R d . By contrast, a remarkable theorem of Bárány and Matoušek states that the fractional Helly number of convex lattice sets drops back down to $$d+1$$ d + 1 , matching the classical fractional Helly theorem of Katchalski and Liu. In this paper we generalize the Bárány–Matoušek 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 $$2^{r}$$ 2 r . This bound is nearly tight, as illustrated by the case of box convexity in $$\mathbb {R}^d$$ R d , whose Radon number is $$\Theta (\log d)$$ Θ ( log d ) and fractional Helly number equals $$d+1$$ d + 1 .