Uncentered Blaschke-Santaló inequalities for the Gaussian measure
Abstract
We study the maximizers of the generalized volume product γσn(A) γσn(A∘) among all measurable subsets A⊂ℝn, where A∘ denotes the polar set of A, and where γσn denotes the centered Gaussian probability measure on ℝn with covariance σ2 In, σ>0. It turns out that the maximizers depend on σ. We prove that they exist and are convex bodies. In dimension n=1, we find the exact form of the maximizers. In dimension n≥ 2, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for σ2 ≤ 1/n we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for σ2≥ 2/(n+1). In dimension n=2, we close the gap by showing that the Euclidean unit ball is the unique maximizer for σ2 ≤ 2/3.
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BibTeX
@misc{artsteinavidan2026uncentered,
title = {{Uncentered Blaschke-Santaló inequalities for the Gaussian measure}},
author = {S. Artstein-Avidan and M. Fradelizi and K. Wyczesany},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18472},
eprint = {2609.18472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18472},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uncentered Blaschke-Santaló inequalities for the Gaussian measure
AU - S. Artstein-Avidan
AU - M. Fradelizi
AU - K. Wyczesany
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18472
UR - https://arxiv.org/abs/2609.18472
AB - We study the maximizers of the generalized volume product γ_σ^n(A) γ_σ^n(A^∘) among all measurable subsets A⊂ℝ^n, where A^∘ denotes the polar set of A, and where γ_σ^n denotes the centered Gaussian probability measure on ℝ^n with covariance σ^2 I_n, σ>0. It turns out that the maximizers depend on σ. We prove that they exist and are convex bodies. In dimension n=1, we find the exact form of the maximizers. In dimension n≥ 2, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for σ^2 ≤ 1/n we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for σ^2≥ 2/(n+1). In dimension n=2, we close the gap by showing that the Euclidean unit ball is the unique maximizer for σ^2 ≤ 2/3.
ER -