On translates of convex measures

The following alternative is proved for a convex Radon measure $\mu$, on a locally convex space $X$ and for an arbitrary direction $h\in X$: either $\mu$ is differentiable in the direction $h$ in the sense of Skorokhod and $\|\mu _h-\mu \|\geqslant 2-2e^{-\frac 12\|d_h\mu \|}$, or $\mu$ and $\mu _{th}$ are mutually singular for all $t\in \mathbb R\setminus \{0\}$.