A sharp lower bound for torsional rigidity of convex domains

For every bounded convex domain $Ω\subset\mathbb R^d$, $d\ge2$, we establish the following sharp inequality for torsional rigidity $$T(Ω)\ge\frac{2(d+1)}{d}\int\limits_Ω\operatorname{dist}(x,\partialΩ)^2\dd x,$$ where $T(Ω)=4\int\limits_Ωu\dd x$ and $-Δu=1$ in $Ω$, $u\in H^1_0(Ω)$. Equality holds for balls.

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-10-29
DOI
https://doi.org/10.48550/arxiv.2610.39666
Primary Topic
Mathematical Inequalities and Applications
Type
preprint
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preprint

A sharp lower bound for torsional rigidity of convex domains

M. A. Perelmuter
arXiv (Cornell University)
Mathematical Inequalities and Applications
preprint

A sharp lower bound for torsional rigidity of convex domains

M. A. Perelmuter
preprint en

Abstract

For every bounded convex domain $Ω\subset\mathbb R^d$, $d\ge2$, we establish the following sharp inequality for torsional rigidity $$T(Ω)\ge\frac{2(d+1)}{d}\int\limits_Ω\operatorname{dist}(x,\partialΩ)^2\dd x,$$ where $T(Ω)=4\int\limits_Ωu\dd x$ and $-Δu=1$ in $Ω$, $u\in H^1_0(Ω)$. Equality holds for balls.

arXiv (Cornell University)
Mathematical Inequalities and Applications
DOIOpen Access PDFOpenAlex
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A sharp lower bound for torsional rigidity of convex domains — M. A. Perelmuter · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS