Extension of MARRMs to closed convex cones while preserving convexity and lower semicontinuity

Ascertain conditions under which a multi-asset return risk measure η_R defined on a model set C ⊂ L^0_{++} can be extended to a proper, convex, and lower semicontinuous function on a closed convex cone containing C.

Background

In discussing direct convex duality for MARRMs on C, the authors identify that when C ⊂ L^1 the set C is not a closed convex cone in standard L^p topologies, which obstructs standard Fenchel-Moreau arguments. Even if η_R is lower semicontinuous on C, it is unclear under what assumptions it admits an extension to the norm-closure of C that retains lower semicontinuity and convexity.

They explicitly pose the problem of finding structural conditions that ensure such an extension exists—proper, convex, and lower semicontinuous—on a closed convex cone containing C.