The Gaussian Brunn–Minkowski conjecture for convex sets containing the origin

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, defined by

$$\mathrm{d}\gamma_n(x)=\frac{1}{(2\pi)^{n/2}}e^{-\frac{|x|^2}{2}}\,\mathrm{d}x.$$

For $\lambda\in(0,1)$ and sets $A,B\subset\mathbb{R}^n$, write $(1-\lambda)A+\lambda B=\{(1-\lambda)a+\lambda b:a\in A,\ b\in B\}$. Gaussian Brunn–Minkowski conjecture. The inequality

$$\gamma_n\bigl((1-\lambda)A+\lambda B\bigr)^{1/n}\geq(1-\lambda)\gamma_n(A)^{1/n}+\lambda\gamma_n(B)^{1/n}$$

holds for any closed convex sets $A,B$ such that $0\in A\cap B$. This conjecture proposes a Brunn–Minkowski inequality in Gaussian space under the natural condition that both convex sets contain the origin; without such a positional condition, the inequality fails in general, for example for a Euclidean ball and a sufficiently distant translate.