D. Gerbner

Separating families of convex sets

Two elements, $x$ and $y$, are separated by a set $S$ if it contains exactly one of $x$ and $y$. We prove that any set of $n$ points in general position in the plane can be separated by $O(n\log\log n/\log n)$ convex sets, and for some point sets $Ω(n/\log n)$ convex sets are necessary.