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Livshyts","submitted_at":"2021-06-30T20:45:49Z","abstract_excerpt":"We show that for any log-concave measure $\\mu$ on $\\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\\lambda\\in [0,1],$ $$\\mu((1-\\lambda) K+\\lambda L)^{c_n}\\geq (1-\\lambda) \\mu(K)^{c_n}+\\lambda\\mu(L)^{c_n},$$ where $c_n\\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \\cite{GZ}, Colesanti, L, Marsiglietti \\cite{CLM}). 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