Concentration of the mixed discriminant of well-conditioned matrices

We call an n-tuple Q1,…,Qn of positive definite n×n real matrices α-conditioned for some α≥1 if for the corresponding quadratic forms qi:Rn⟶R we have qi(x)≤αqi(y) for any two vectors x,y∈Rn of Euclidean unit length and qi(x)≤αqj(x) for all 1≤i,j≤n and all x∈Rn. An n-tuple is called doubly stochastic if the sum of Qi is the identity matrix and the trace of each Qi is 1. We prove that for any fixed α≥1 the mixed discriminant of an α-conditioned doubly stochastic n-tuple is nO(1)e−n. As a corollary, for any α≥1 fixed in advance, we obtain a polynomial time algorithm approximating the mixed discriminant of an α-conditioned n-tuple within a polynomial in n factor.