Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Stephan Weis

Czech Technical University, Faculty of Electrical Engineering, Prague, Czech Republic

maths@weis-stephan.de

Keywords

  • Convex set
  • extreme set
  • face
  • face generated by a point
  • relative interior
  • relative algebraic interior
  • Radon-Nikodym derivative
  • convex core

Mathematics Subject Classification

52A05, 46E27

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Suggested citation

S. Weis. “A Note on Faces of Convex Sets.” Journal of Convex Analysis 32 (2025), No. 3, 901–919. https://doi.org/10.68381/jca32046

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.