We give an improved algorithm for weighted directed planar diameter that runs in $\widetilde{O}(n^{3/2})$ time. This improves upon the $\widetilde{O}(n^{5/3})$ time algorithm for weighted directed planar diameter of Gawrychowski, Kaplan, Mozes, Sharir, and Weimann [SIAM J. Comput. 2021] and the recent $\widetilde{O}(n^{8/5})$ time algorithm for unweighted undirected planar diameter of Fredslund-Hansen, Mozes, and Weimann [SOSA 2027]. The main new result is an improved data structure for computing the farthest vertex when given feasible planar distances to the vertices on the outer face. The critical insight is the fact that normalized distances to the vertices of a face behave like pseudolines, for which we can leverage range searching and upper envelope data structures. Using the farthest vertex data structure, we are able to compute all vertex eccentricities and diameter in $\widetilde{O}(n^{3/2})$.
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.