Proper faces of convex sets in arbitrary real vector spaces are characterized by generalized semispaces, compatible total preorders, and regular step-affine functions; all three descriptions are equivalent.
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Characterizations of Faces of Convex Sets in Infinite-dimensional Vector Spaces
Proper faces of convex sets in arbitrary real vector spaces are characterized by generalized semispaces, compatible total preorders, and regular step-affine functions; all three descriptions are equivalent.