We will show that for any even log-concave measure \mu and any pair of symmetric convex sets K and L, and any t between 0 and 1, one has the inequality: \mu(tK+(1-t)L)^{c(n)}\geq t\mu(K)^{c(n)}+(1-t)\mu(L)^{c(n)}, Where c(n)=n^{-4-o(1)}. This constitutes progress towards the Dimensional Brunn-Minkowski conjecture.