Abstract

We show that, given a closed convex set KK containing the origin in its interior, the support function of the set {y∈K∗∣∃x∈K such that ⟨x,y⟩=1}\{y\in K^* \mid \exists x\in K\text{ such that } \langle x,y \rangle =1\} is the pointwise smallest among all sublinear functions σ\sigma such that K={x∣σ(x)≤1}K=\{x \mid \sigma(x)\leq 1\}

Author information

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

Amitabh Basu

Dept. of Mathematics, University of California, One Shields Avenue, Davis, CA 95616, U.S.A.

abasu@math.ucdavis.edu

Gérard Cornuéjols

Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 12180, U.S.A.

gc0v@andrew.cmu.edu

Giacomo Zambelli

Dept. of Management, London School of Economics, Houghton Street, London WC2A 2AE, England

g.zambelli@lse.ac.uk

References

  1. A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli: Maximal lattice-free convex sets in linear subspaces, Math. Oper. Res. 35 (2010) 704–710.https://doi.org/10.1287/moor.1100.0461Google Scholar
  2. A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli: Minimal inequalities for an infinite relaxation of integer programming, SIAM J. Discrete Math. 24 (2010) 158–168.https://doi.org/10.1137/090756375Google Scholar
  3. S. S. Dey, L. A. Wolsey: Constrained infinite group relaxations of MIPs, SIAM J. Optim. 20 (2010) 2890–2912.https://doi.org/10.1137/090754388Google Scholar
  4. E. L. Johnson: Characterization of facets for multiple right-hand side choice linear programs, Math. Program. Study 14 (1981) 112–142.https://doi.org/10.1007/BFb0120925Google Scholar
  5. J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-56468-0Google Scholar
  6. L. Hörmander: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Ark. Mat. 3 (1954) 181–186.https://doi.org/10.1007/BF02589354Google Scholar
  7. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1969).Google Scholar
  8. S. Straszewicz: Über exponierte Punkte abgeschlossener Punktemengen, Fundam. Math. 24 (1935) 139–143.https://doi.org/10.4064/fm-24-1-139-143Google Scholar

Suggested citation

A. Basu, G. Cornuéjols, G. Zambelli. “Convex Sets and Minimal Sublinear Functions.” Journal of Convex Analysis 18 (2011), No. 2, 427–432. https://doi.org/10.68381/jca18026

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.