Let $P$ be a finite point set in $\mathbb{R}^d$. A weak $\varepsilon$-net for $P$ with respect to convex sets is a finite set $X\subset\mathbb{R}^d$ intersecting every convex set containing at least $\varepsilon|P|$ points of $P$. Determining their optimal worst-case size was described by Alon, Kalai, Matousek, and Meshulam [Adv. Appl. Math., 2002] as "one of the truly important open problems in combinatorial geometry." We give elementary constructions which establish the first polynomially superlinear lower bounds in fixed dimension. Already in $\mathbb{R}^3$, we construct point sets for which every weak $\varepsilon$-net has size $Ω(\varepsilon^{-3/2})$. More generally, for every fixed $d\ge3$, we obtain a lower bound of $Ω_d(\varepsilon^{-γ_d})$, where $γ_d=\sqrt{d/2}+O(1)$. Consequently, no upper bound of the form $O_d(\varepsilon^{-C})$ can hold with an absolute constant $C$ which is independent of the dimension. Despite the recent progress on upper bounds, the strongest previous general lower bound remained $Ω_d(\varepsilon^{-1}\log^{d-1}(1/\varepsilon))$, due to Bukh, Matousek, and Nivasch [Israel J. Math., 2011]. To obtain our results, we apply a quadratic lift to grid-based configurations with many point-line incidences. The bounds also yield polynomial separations between fractional and integral piercing numbers of convex families, and lower bounds for Hadwiger-Debrunner numbers.

翻译：暂无翻译