 Solution

For $p\in {\mathcal{U}}^{\ast }$, the definition of the convex conjugate giveswhere the substitution $w=u-v$ is a bijection of the independent pairs. ThereforeThis infimal convolution identity does not need convexity of $E,F$ or attainment of the inner infimum. The assumptions ensure that both functions have nonempty effective domains; their conjugates never take $-\infty$, so the separated sum is well defined, allowing $+\infty$.