By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for infinite dimensions. I am specifically interested in the case where the dimension is uncountable.

Note that this strong convexity is equivalent to the strong convexity of negative entropy function $H(P) = \int dP \log(dP)$.