Let $X$ be a infinite-dimensional Hausdorff real topological vector space and $f:X\to[-\infty, \infty]$ be a proper function. Define the convex envelope of $f$ to be the largest convex minorant of $f$.
My question: are there any results in convex analysis which give sufficient conditions for the convex envelope of $f$ to be lower semicontinuous?
In a finite dimensional space, there are broad cases where convexity implies continuity and hence lower semicontinuity. For example, Proposition 2.17 of Barbu's "Convexity and Optimization in Banach Spaces" states that if $X$ be a finite dimensional Hausdorff topological vector space and $f:X\to[-\infty,\infty]$ is a proper convex function, then $f$ is continuous on the interior of $\text{Dom}(f)$. This is statement that applies to all convex functions, but I only need a statement for the convex envelope.
Are there any analogous results for similar (yet infinite-dimensional) spaces? I am willing to assume $X$ is a Banach or even Hilbert Space if needed.