Abstract

We study touching cones of a (not necessarily closed) convex set in a finite-dimensional real Euclidean vector space and we draw relationships to other concepts in Convex Geometry. Exposed faces correspond to normal cones by an antitone lattice isomorphism. Poonems generalize the former to faces and the latter to touching cones, these extensions are non-isomorphic, though. We study the behavior of these lattices under projections to affine subspaces and intersections with affine subspaces. We prove a theorem that characterizes exposed faces by assumptions about touching cones. For a convex body K the notion of conjugate face adds an isotone lattice isomorphism from the exposed faces of the polar body K∘K^\circ to the normal cones of K. This extends to an isomorphism between faces and touching cones.

Author information

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

Stephan Weis

Department Mathematik, Friedrich-Alexander-Universität, Bismarckstraße 1½, 91054 Erlangen, Germany

weis@mi.uni-erlangen.de

Keywords

  • Convex set
  • exposed face
  • normal cone
  • poonem
  • face
  • touching cone
  • projection
  • intersection

Mathematics Subject Classification

52A10, 52A20, 94A17

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Suggested citation

S. Weis. “A Note on Touching Cones and Faces.” Journal of Convex Analysis 19 (2012), No. 2, 323–353. https://doi.org/10.68381/jca19019

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