On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of $\mathbf{\text{symmetric}}$ convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$\gamma (\lambda K+(1-\lambda )L{)}^{\frac{1}{2n}}\ge \lambda \gamma (K{)}^{\frac{1}{2n}}+(1-\lambda )\gamma (L{)}^{\frac{1}{2n}}\mathrm{.}$$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>0,$ with respect to the addition of symmetric convex sets.