Depth based on translates of a convex body: geometry, floating bodies, and empirical theory

Abstract

For a fixed full-dimensional convex body $K\subset\mathbb R^d$, we study the $K$-depth of a point $y$ with respect to a probability measure $μ$ by comparing the masses of translates of $K$ that fail to contain $y$. Its upper level sets are intersections of high-mass translates of $K$, and hence are convex and admit an exact representation as Minkowski erosions. We distinguish the closed core determined by the maximisers of the translation-mass function from the effective support on which the depth exceeds its unavoidable baseline. We establish continuity, affine equivariance, comparison with halfspace (Tukey) depth, and normal-cone and smoothness criteria. For uniform measures on convex bodies, the regions coincide exactly with the ball floating bodies of Schütt, Werner and Yalikun; for positive continuous densities they coincide with the weighted ball floating bodies of Werner and Yalikun. For dilates $tK$, the depth converges to a depth generated by the tangent cones of $K$ as $t \to \infty$; for smooth $K$ this limit is Tukey's halfspace depth, whereas nonsmooth testing bodies can produce a genuinely different limit. The worst-case maximal $K$-depth is exactly $1/h_T(K)$, where $h_T(K)$ is the Helly number of the family of translates of $K$. Under a uniform interior rolling-ball condition and compact support and having zero mass on hyperplanes, the convergence to halfspace depth is monotone and uniform with an explicit error bound. If the density is bounded by $L$, then this bound is of order $O(t^{-1})$ and this order is sharp. We also record that the dependence on the testing body is not monotone in general, while Hausdorff perturbations of $K$ give stable depth functions and regions at regular levels. Finally, we prove uniform consistency and pointwise weak limits for empirical $K$-depth, including the non-Gaussian limit caused by nonunique active translates.