A Smooth Covariogram and the Transition Between Projection Bodies
Abstract
We introduce a smooth radial approximation of the covariogram of a convex body. The construction replaces the intersection K∩(K+x) by the radial (−p)-mean of K and K+x, producing a family gK,p which converges pointwise to the classical covariogram as p→∞. For each fixed p>0, the function gK,p is smooth near the origin, and its Hessian is computed explicitly in terms of the operator defining the polar L2-projection body. We study the simultaneous limit in which p→∞ and the level deficit is δ=pα, α<0. The normalized level sets exhibit three regimes: the polar projection body appears for α>−1, the polar L2-projection body appears for α<−1, and a critical transition body appears at α=−1. This transition body is described by an explicit logcosh-type boundary integral, placing the critical regime naturally within the framework of Orlicz projection bodies.
Related
- Stability of Hyperplane Projections of the Regular Simplex and Lp-Projection Bodies of balls in ℓqn(ℝ)
- Algebraically Integrable Smooth Convex Bodies
- Geometric desingularisation of the sharp-to-smooth travelling wave transition
- The Petty Conjecture for Convex Bodies of Revolution
- Riemannian convex bodies with a geodesic segment on the boundary
BibTeX
@misc{haddad2026smooth,
title = {{A Smooth Covariogram and the Transition Between Projection Bodies}},
author = {J. Haddad and C. H. Jiménez and M. Resende Guedes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11700},
eprint = {2610.11700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Smooth Covariogram and the Transition Between Projection Bodies
AU - J. Haddad
AU - C. H. Jiménez
AU - M. Resende Guedes
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11700
UR - https://arxiv.org/abs/2610.11700
AB - We introduce a smooth radial approximation of the covariogram of a convex body. The construction replaces the intersection K∩(K+x) by the radial (−p)-mean of K and K+x, producing a family g_K,p which converges pointwise to the classical covariogram as p→∞. For each fixed p>0, the function g_K,p is smooth near the origin, and its Hessian is computed explicitly in terms of the operator defining the polar L_2-projection body. We study the simultaneous limit in which p→∞ and the level deficit is δ=p^α, α<0. The normalized level sets exhibit three regimes: the polar projection body appears for α>−1, the polar L_2-projection body appears for α<−1, and a critical transition body appears at α=−1. This transition body is described by an explicit logcosh-type boundary integral, placing the critical regime naturally within the framework of Orlicz projection bodies.
ER -