Let $\mathcal{P}_��:=\mathcal{P}_{����}$ denote a Poisson point process of intensity $����$ on $[0,1]^d,d\geq2$, with $��$ a bounded density on $[0,1]^d$ and $��\in(0,\infty)$. Given a closed subset $\mathcal{M}\subset[0,1]^d$ of Hausdorff dimension $(d-1)$, we consider general statistics $\sum_{x\in\mathcal{P}_��}��(x,\mathcal{P} _��,\mathcal{M})$, where the score function $��$ vanishes unless the input $x$ is close to $\mathcal{M}$ and where $��$ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics $\sum_{x\in\mathcal{ P}_��}��(��^{1/d}x,��^{1/d}\mathcal{P}_��,��^{1/d}\mathcal{M})$ as $��\to\infty$. When $\mathcal{M}$ is of class $C^2$, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order $\mathrm{Vol}(��^{1/d}\mathcal{M})$. We use the general results to deduce variance asymptotics and central limit theorems for statistics arising in stochastic geometry, including Poisson-Voronoi volume and surface area estimators, answering questions in Heveling and Reitzner [Ann. Appl. Probab. 19 (2009) 719-736] and Reitzner, Spodarev and Zaporozhets [Adv. in Appl. Probab. 44 (2012) 938-953]. The general results also yield the limit theory for the number of maximal points in a sample.

Published in at http://dx.doi.org/10.1214/13-AAP992 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)