Let $A$ be a closed, bounded, and convex subset of the Banach space $X.$ Suppose $V$ is a convex and open subset of $X$ containing $A,$ $( A \subset V).$ Is there an open neighborhood of zero, say $U$ in $X$ such that $A+U \subseteq V$ ?
I know that When $A$ is weakly compact, the answer is yes, but general case is unclear to me!
If X is reflexive then the answer is yes (based on your argument in the case of weakly compact A). In fact, if X is reflexive then the closed unit ball of X is weakly compact. Since, A is bounded, it is contained in rBall(X) for some r>0. A is also weakly closed since it is norm closed and convex. Thus A is weakly compact. Now use your result on weakly compact case.