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  1. We show that for any even log-concave probability measureμ<#comment/>\muonRn\mathbb {R}^n, any pair of symmetric convex setsKKandLL, and anyλ<#comment/>∈<#comment/>[0,1]\lambda \in [0,1],μ<#comment/>((1−<#comment/>λ<#comment/>)K+λ<#comment/>L)cn≥<#comment/>(1−<#comment/>λ<#comment/>)μ<#comment/>(K)cn+λ<#comment/>μ<#comment/>(L)cn,\begin{equation*} \mu ((1-\lambda ) K+\lambda L)^{c_n}\geq (1-\lambda ) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*}wherecn≥<#comment/>n−<#comment/>4−<#comment/>o(1)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.

     
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