Convex order under majorization for projections of ell-p balls
Abstract
Let n≥1 and 1≤ p≤2, and let X be uniformly distributed on the unit ball Bpn of ℓpn. We prove that if (a12,…,an2) majorizes (b12,…,bn2), then ⟨ b,X⟩2 is dominated by ⟨ a,X⟩2 in convex order. For p<2 this gives a nontrivial comparison of the complete squared-projection laws, whereas at p=2 it reduces to equality by rotational invariance. The result strengthens known majorization inequalities for individual power moments and provides a single distributional source for several familiar projection inequalities. Its proof combines a two-dimensional hinge-function comparison with a preservation theorem for Beta--Rademacher perturbations, which lifts elementary majorization transfers through the Dirichlet representation of the uniform measure on Bpn. The comparison is stable under a common independent radial scaling and therefore extends to ℓp-radial measures with nonincreasing radial density. Consequences include simultaneous inequalities for moments, hinge functions, and Laplace transforms, together with Schur-convexity of Gaussian-regularized central-section densities. The range is sharp in the stated direction: for p>2, a support obstruction already occurs between a coordinate direction and a two-coordinate direction.
Related
- The Fekete-Szegö problem for starlike mappings of order α and quasi-convex mappings of type B in the unit balls of complex Banach spaces
- Stability of Hyperplane Projections of the Regular Simplex and Lp-Projection Bodies of balls in ℓqn(ℝ)
- Busemann G-spaces with convex balls
- Free boundary minimal annuli in convex balls
- On asymptotically conical curve shortening flows with convex projections
BibTeX
@misc{fafe2026convex,
title = {{Convex order under majorization for projections of ell-p balls}},
author = {Soufiane Fafe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.11451},
eprint = {2605.11451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.11451},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convex order under majorization for projections of ell-p balls
AU - Soufiane Fafe
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2605.11451
UR - https://arxiv.org/abs/2605.11451
AB - Let n≥1 and 1≤ p≤2, and let X be uniformly distributed on the unit ball B_p^n of ℓ_p^n. We prove that if (a_1^2,…,a_n^2) majorizes (b_1^2,…,b_n^2), then ⟨ b,X⟩^2 is dominated by ⟨ a,X⟩^2 in convex order. For p<2 this gives a nontrivial comparison of the complete squared-projection laws, whereas at p=2 it reduces to equality by rotational invariance. The result strengthens known majorization inequalities for individual power moments and provides a single distributional source for several familiar projection inequalities. Its proof combines a two-dimensional hinge-function comparison with a preservation theorem for Beta--Rademacher perturbations, which lifts elementary majorization transfers through the Dirichlet representation of the uniform measure on B_p^n. The comparison is stable under a common independent radial scaling and therefore extends to ℓ_p-radial measures with nonincreasing radial density. Consequences include simultaneous inequalities for moments, hinge functions, and Laplace transforms, together with Schur-convexity of Gaussian-regularized central-section densities. The range is sharp in the stated direction: for p>2, a support obstruction already occurs between a coordinate direction and a two-coordinate direction.
ER -