Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

Let $(Phi,Psi)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^Phi$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^Phi$ characterizes closedness with respect to the topology $sigma(L^Phi,L^Psi)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^Phi$, order closedness and $sigma(L^Phi,L^Psi)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $sigma(L^Phi,L^Psi)$-closedness of convex sets in $L^Phi$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $sigma(L^Phi,L^Psi)$; that is, a convex set is $sigma(L^Phi,L^Psi)$-closed if and only if it is closed with respect to the bounded-$sigma(L^Phi,L^Psi)$ topology. As a result, we show that order closedness and $sigma(L^Phi,L^Psi)$-closedness of convex sets in $L^Phi$ are equivalent if and only if either $Phi$ or $Psi$ satisfies the $Delta_2$-condition. Using this, we prove the surprising result that: emph{If (and only if) $Phi$ and $Psi$ both fail the $Delta_2$-condition, then there exists a coherent risk measure on $L^Phi$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^Phi, L^Psi)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^Phi,H^Psi)$.