Abstract

Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of non-exposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One of them assigns conjugate faces between the convex bodies. The second and third Galois connection is defined between the touching cones and the faces of each convex body separately. While the former is well-known, we introduce the latter in this article for any convex set in any finite dimension. We demonstrate our results about conjugate faces with planar convex bodies and planar self-dual convex bodies, for which we also include constructions.

Author information

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

Stephan Weis

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

sweis@mis.mpg.de

Keywords

  • Convex body
  • polar
  • conjugate face
  • non-exposed point
  • singular face
  • dual
  • self-dual

Mathematics Subject Classification

52A10, 52A20

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Suggested citation

S. Weis. “Duality of Non-Exposed Faces.” Journal of Convex Analysis 19 (2012), No. 3, 815–835. https://doi.org/10.68381/jca19042

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.