Depth based on translates of a convex body: geometry, floating bodies, and empirical theory
Abstract
For a fixed full-dimensional convex body K⊂ℝd, 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 → ∞; 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/hT(K), where hT(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.
Related
- Union of Translates of a Convex Body
- The Petty Conjecture for Convex Bodies of Revolution
- Riemannian convex bodies with a geodesic segment on the boundary
- Continuity and monotonicity of weighted perimeters of convex bodies
- Comment on "The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank"
BibTeX
@misc{molchanov2026depth,
title = {{Depth based on translates of a convex body: geometry, floating bodies, and empirical theory}},
author = {Ilya Molchanov and Stanislav Nagy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26108},
eprint = {2610.26108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26108},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Depth based on translates of a convex body: geometry, floating bodies, and empirical theory
AU - Ilya Molchanov
AU - Stanislav Nagy
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26108
UR - https://arxiv.org/abs/2610.26108
AB - For a fixed full-dimensional convex body K⊂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 → ∞; 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.
ER -