A sharp Brégman–Minc bound for permanents
We prove a sharp Brégman–Minc bound for permanents of nonnegative matrices that was conjectured by Samorodnitsky and Gurvits. Specifically, for $\phi(x)=\Gamma(1+1/x)^{-x}$, with $\phi(0)=0$, applied entrywise, we show that every nonnegative row-stochastic square matrix $A$ satisfies $\operatorname{per}\phi[A]\le1$, which is the Brégman–Minc inequality when each row of $A$ is uniform on its support. The crux of our proof is a scalar inequality on the simplex, which we prove using a carefully chosen yet elementary integral representation. We note brief applications to the entropy of a random…
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