The Brunn-Minkowski inequality is one of the deepest results in convex geometry: the volume of the Minkowski sum of two convex bodies is at least the sum of their individual volumes raised to appropriate powers. Analogous inequalities hold for other geometric functionals — eigenvalues, capacities, torsional rigidities — and the hope was that they extend to the Ornstein-Uhlenbeck setting, where Gaussian measure replaces Lebesgue measure.
This paper shows they do not. For the torsional rigidity of the Ornstein-Uhlenbeck operator on general bounded convex sets, neither concavity nor convexity holds. For centrally symmetric sets, the expected fractional power law also fails. The geometric inequalities that hold in the Euclidean world break when the background measure has Gaussian decay.
The positive result is narrow: for Euclidean balls centered at the origin, the cube root of the Gaussian torsional rigidity is convex under Minkowski addition. But this is the only clean statement that survives. For the first eigenvalue, even centrally symmetric sets provide counterexamples.
The results answer questions posed by Cordero-Erausquin, Eskenazis, and Colesanti et al. — and they answer them negatively. The conjectured inequalities are simply false.
The pattern is instructive: Euclidean geometry has symmetries that make many inequalities “work” — translation invariance, scaling homogeneity. The Gaussian setting breaks these symmetries. Functionals that were well-behaved under Minkowski addition in flat space become unruly when the underlying measure penalizes distance from the origin. The geometry of the measure fights the geometry of the sets.