The fractional Helly number for separable convexity spaces

A convex lattice set in ${\mathbb{Z}}^d$ is the intersection of a convex set in ${\mathbb{R}}^d$ and the integer lattice ${\mathbb{Z}}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals $2^d$, while a remarkable theorem of Bárány and Matoušek states that the fractional Helly number is only $d+1$. In this paper we generalize their result to abstract convexity spaces that are equipped with a suitable separation property. We also disprove a conjecture of Bárány and Kalai about an existence of fractional Helly property for a family of solutions of bounded-degree polynomial inequalities.