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A note on faces of convex sets

The faces of a convex set owe their relevance to an interplay between convexity and topology that is systematically studied in the work of Rockafellar. Infinite-dimensional convex sets are excluded from this theory as their relative interiors may be empty. Shirokov and the present author answered this issue by proving that every point in a convex set lies in the relative algebraic interior of the face it generates. This theorem is proved here in a simpler way, connecting ideas scattered throughout the literature. This article summarizes and develops methods for faces and their relative algebraic interiors and applies them to spaces of probability measures.