A Note on Testing Intersection of Convex Sets in Sublinear Time

We present a simple sublinear time algorithm for testing the following geometric property. Let $P_1, ..., P_n$ be $n$ convex sets in $\mathbb{R}^d$ ($n \gg d$), such as polytopes, balls, etc. We assume that the complexity of each set depends only on $d$ (and not on the number of sets $n$). We test the property that there exists a common point in all sets, i.e. that their intersection is nonempty. Our goal is to distinguish between the case where the intersection is nonempty, and the case where even after removing many of the sets the intersection is empty. In particular, our algorithm returns PASS if all of the $n$ sets intersect, and returns FAIL with probability at least $1-\epsilon$ if no point belongs to $\frac{\alpha}{d+1} n$ sets, for any given $0 < \alpha, \epsilon < 1$.