Gaussian torsional rigidity exponent conjecture for centrally symmetric convex sets

From papers

Let Ω0\Omega_0 and Ω1\Omega_1 be open, bounded, centrally symmetric subsets of Rn\mathbb{R}^n, and let t[0,1]t\in[0,1]. Write

Ωt=(1t)Ω0+tΩ1.\Omega_t=(1-t)\Omega_0+t\Omega_1.

The Gaussian torsional rigidity of a set Ω\Omega is denoted by Tγ(Ω)T_\gamma(\Omega). Gaussian torsional rigidity exponent conjecture. One should have

Tγ(Ωt)1/3(1t)Tγ(Ω0)1/3+tTγ(Ω1)1/3.T_\gamma(\Omega_t)^{1/3}\leq (1-t)T_\gamma(\Omega_0)^{1/3}+tT_\gamma(\Omega_1)^{1/3}.

Equality should hold if and only if Ω0=Ω1\Omega_0=\Omega_1. This conjecture extends the proved ball case to centrally symmetric convex sets; the source gives no resolution for the general case.

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Sources & referencesView supporting material

Primary source

Francisco Marín Sola and Francesco Salerno, “Remarks on Brunn-Minkowski-type inequalities related to the Ornstein-Uhlenbeck operator”, arXiv:2603.19164 (2026).

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