Cramér transform, half-space depth and threshold phenomena for convex bodies
Abstract
We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure μK on a convex body K⊆ℝn, we prove the sharp pointwise comparison ΛK*(x)≤ −log qK(x)≤ ΛK*(x)+1/2log n+C, x∈int(K), where C is an absolute constant. The order log n is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cramér transform. As consequences, we obtain sharp-order moment and tail estimates for ΛK* and identify exp(ΛK*(x)), up to polynomial factors in the dimension, with the number of independent samples needed for x to be captured by their random convex hull. We also establish an O(n2) variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on ℓp-balls for every p>1. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.
Related
- The Laplace transform of the time of ruin in the Cramér--Lundberg model with Erlang E2(μ) claims: spectrum, exact moments and matrix reduction
- 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{pafis2026cramér,
title = {{Cramér transform, half-space depth and threshold phenomena for convex bodies}},
author = {Minas Pafis},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29972},
eprint = {2608.29972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29972},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cramér transform, half-space depth and threshold phenomena for convex bodies
AU - Minas Pafis
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29972
UR - https://arxiv.org/abs/2608.29972
AB - We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure μ_K on a convex body K⊆ℝ^n, we prove the sharp pointwise comparison Λ_K^*(x)≤ −log q_K(x)≤ Λ_K^*(x)+1/2log n+C, x∈int(K), where C is an absolute constant. The order log n is optimal, as shown by the Euclidean ball. The proof combines exponential tilting, one-dimensional log-concavity, and self-concordance of the Cramér transform. As consequences, we obtain sharp-order moment and tail estimates for Λ_K^* and identify exp(Λ_K^*(x)), up to polynomial factors in the dimension, with the number of independent samples needed for x to be captured by their random convex hull. We also establish an O(n^2) variance bound for the logarithmic half-space depth and use it to derive a general criterion for sharp thresholds of random convex hulls. In particular, this criterion applies to the uniform measures on ℓ_p-balls for every p>1. These results establish a quantitative link between large-deviation cost, geometric depth, and sampling complexity in high-dimensional convex geometry.
ER -