Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity
Abstract
Let u be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain Ω⊂ ℝn, i.e., the solution of Δu − x · ∇ u = −1 in Ω with u = 0 on ∂Ω. Let Tγ(Ω) = ∫Ωu dγ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality Tγ((1−t)Ω0 + tΩ1)α ≤ (1−t) Tγ(Ω0)α + t Tγ(Ω1)α fails for every exponent α> 0 and in every dimension n ≥ 2, for a pair of convex bodies centrally symmetric with respect to the origin, which may be taken smooth with positive curvature. This answers Conjecture 1.4 for any n ≥ 2, and Question~(Q), of Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for Tγ at a ball Ω0 under Minkowski perturbations. When the perturbing body Ω1 has small torsion and large mean width, Tγ((1−t)Ω0 + tΩ1) increases to first order. Since the Minkowski combination has larger torsion than both endpoints, no exponent can repair the inequality. For n=1, convexity with the optimal exponent 1/3 holds on symmetric intervals by results of the same authors, \cite{MSS26}. We prove that the logarithm of the torsion is neither convex nor concave along Minkowski combinations of symmetric intervals, that no non-zero exponent yields concavity, and that convexity fails for every positive exponent when one set is a union of two intervals or when the sets are reflected off-center intervals.
Related
- Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality
- A sharp lower bound for torsional rigidity of convex domains
- Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons
- Computer-assisted local maximality of regular polygons for torsional rigidity
- Sharp Bounds on Torsional Rigidity of Metric Graphs with Signed Delta-Vertex Strengths
BibTeX
@misc{nguyen2026failure,
title = {{Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity}},
author = {Xuan Hien Nguyen and Alina Stancu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29941},
eprint = {2608.29941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29941},
}
RIS
TY - UNPB
TI - Failure of a Brunn-Minkowski-type inequality for the Gaussian torsional rigidity
AU - Xuan Hien Nguyen
AU - Alina Stancu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.29941
UR - https://arxiv.org/abs/2608.29941
AB - Let u be the torsion function for the Ornstein-Uhlenbeck operator on a bounded domain Ω⊂ ℝ^n, i.e., the solution of Δu − x · ∇ u = −1 in Ω with u = 0 on ∂Ω. Let T_γ(Ω) = ∫_Ωu dγ be the Gaussian torsional rigidity. We prove that the Brunn-Minkowski-type inequality T_γ((1−t)Ω_0 + tΩ_1)^α ≤ (1−t) T_γ(Ω_0)^α + t T_γ(Ω_1)^α fails for every exponent α> 0 and in every dimension n ≥ 2, for a pair of convex bodies centrally symmetric with respect to the origin, which may be taken smooth with positive curvature. This answers Conjecture 1.4 for any n ≥ 2, and Question~(Q), of Marín Sola and Salerno in the negative. The mechanism is a first-order lower bound for T_γ at a ball Ω_0 under Minkowski perturbations. When the perturbing body Ω_1 has small torsion and large mean width, T_γ((1−t)Ω_0 + tΩ_1) increases to first order. Since the Minkowski combination has larger torsion than both endpoints, no exponent can repair the inequality. For n=1, convexity with the optimal exponent 1/3 holds on symmetric intervals by results of the same authors, \cite{MSS26}. We prove that the logarithm of the torsion is neither convex nor concave along Minkowski combinations of symmetric intervals, that no non-zero exponent yields concavity, and that convexity fails for every positive exponent when one set is a union of two intervals or when the sets are reflected off-center intervals.
ER -