The intrinsic core and minimal faces of convex sets in general vector spaces

Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core, cover much of the folklore, review relevant recent results and present examples illustrating some of the phenomena specific to the facial structure of infinite-dimensional sets.