A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures.

We show that for any even log-concave probability measure \mu on \mathbb {R}^n, any pair of symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} \mu ((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*} where c_n\geq n^{-4-o(1)}. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. [ABSTRACT FROM AUTHOR]