Maximal volume of convex bodies avoiding random points and lacunary dilates

Abstract

For a positive real lacunary sequence and any probability measure $\mu$ with polynomial Fourier decay, we determine, for $\mu$-almost every $x$, the asymptotic maximal volume of convex bodies avoiding the first $N$ dilates of $x$ modulo ${\mathbb{Z}}^d$. Among homothets of a fixed convex body, the maximal volume is asymptotic to $(\log N)/N$. When all convex bodies in the unit cube are allowed, it is asymptotic to $d(\log N)/N$. For independent uniform points in a fixed convex body $\Omega \subset {\mathbb{R}}^d$, the maximal volume of an empty convex body is asymptotic to $d\mathrm{vol}(\Omega )(\log N)/N$ almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.

40 pages