I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ with the property that functions of the form \begin{equation} t \mapsto \mu( A + t h ) \end{equation} are differentiable (in some suitable sense) for measurable $A$ and some $h \in X$.
In the section on log-concave measures, the following statement is written:
We do not know whether any [sic] log-concave measure with infinite-dimensional support has a nonzero direction of differentiability.
Since the review is more than 25 years old: Has there been some progress on this? Note that the case of finite-dimensional support is settled with Theorem 3.3.8 in the cited review paper (all directions are directions of differentiability).
