1. Given a (Hausdorff separated) locally convex space $X$ what can we say about a proper convex function $f:X\to\mathbb{R}$ whose domain $\emptyset\neq D(f):=\{x\in X\mid f(x)<+\infty\}$ has a non-empty (topological) interior?

Recall that when $X$ is barreled the non-empty domain interior spells that the function is continuous on the interior of its domain. So what can happen when the space is not barreled other than continuity?

My take so far in this is to consider the simplest type of a convex function: a sublinear function (positively homogeneous and subadditive). I took $B\subset X^*$ (the topological dual) and $f=\sigma_B$ the support function of $B$; $\sigma_B(x):=\sup_{x^*\in B}x^*(x)$, $x\in D(\sigma_B)$ (the barrier cone of $B$). My reformulation of the problem in this case is:

  1. What can one say about a set whose barrier cone has a non-empty interior?

To simplify even more, in case $0\in {\rm core}\ {D(\sigma_B)}$ (core denotes here the algebraic interior) we get that $D(\sigma_B)=X$ because $D(\sigma_B)$ is an absorbing cone. We find that $B$ is (weak-star) bounded.


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