This seems not to be the usual definition of centrally symmetric (which should be $C = -C$).

Can you clarify what you mean by "bounded" in "bounded measurable convex function"?

@Akira: This set is just the intersection of $\{x \in X \mid g(x) > n\}$ over $n \in \mathbb{N}$.

A copy of this article is available at the homepage of the author: gaborpataki.web.unc.edu/wp-content/uploads/sites/14119/2018/‌​07/…

I would like to add that some assumption on $X$ is necessary. Indeed, if we consider an infinite-dimensional Hilbert space $H$ equipped with its weak topology, then $f = \|\cdot\|$ is convex, lower semicontinuous but not continuous.

Note that the countability of the union also allows for the construction of a choice function: You can choose this element of $A_n$, which comes first in the enumeration of all of the elements of the union.

It seems that the choice $n(i) \equiv \{1,\ldots,p\}$ is always possible? But maybe I do not understand the definition of $a_i^*(s_{n(i)})$.

If $Y$ is a non-reflexive Banach space and $X = Y^*$, then the unit ball $B_Y$ of $Y$ is closed, bounded and convex, but not weak*-closed in $X^* = Y^{**}$ (due to Goldstine theorem). Thus, your first parenthesis might fail and, similarly, $f$ might fail to be weak*-lower semicontinuous.

Sorry, but I don't know. I currently do not have the book available.