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 R-n, any pair of symmetric convex sets K and L, and any lambda is an element of [0, 1], mu((1-lambda)K +lambda L)(cn) >= (1-lambda)mu(K)(cn) +lambda mu(L)(cn), where c(n) >= 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 logconcave measures.