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Computational and Analytical Mathematics pp 161–218Cite as

Compactness, Optimality, and Risk

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

Abstract

This is a survey about one of the most important achievements in optimization in Banach space theory, namely, James’ weak compactness theorem, its relatives, and its applications. We present here a good number of topics related to James’ weak compactness theorem and try to keep the technicalities needed as simple as possible: Simons’ inequality is our preferred tool. Besides the expected applications to measures of weak noncompactness, compactness with respect to boundaries, size of sets of norm-attaining functionals, etc., we also exhibit other very recent developments in the area. In particular we deal with functions and their level sets to study a new Simons’ inequality on unbounded sets that appear as the epigraph of some fixed function f. Applications to variational problems for f and to risk measures associated with its Fenchel conjugate f ∗ are studied.

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Acknowledgements

To finish our contribution let us remark we are very grateful to the anonymous referee who highly improved the redaction of our paper.

B. Cascales and J. Orihuela’s research was partially supported by MTM2008-05396 and MTM2011-25377/MTM Fondos FEDER; Fundación Sénaca 08848/PI/08, CARM; and that of M. Ruiz Galán by Junta de Andalucía grant FQM359.

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Authors and Affiliations

  1. Department of Mathematics, University of Murcia, Campus Espinardo 15, 30100, Murcia, Spain

    B. Cascales & J. Orihuela

  2. Department of Applied Mathematics, E. T. S. Ingeniería. Edificación., c/ Severo Ochoa s/n, 1871, Granada, Spain

    M. Ruiz Galán

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  1. B. Cascales
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  2. J. Orihuela
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  3. M. Ruiz Galán
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Correspondence to B. Cascales .

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Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

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Dedicated to Jonathan Borwein on the occasion of his 60th birthday

Communicated By Jon D. Vanderwerff.

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Cascales, B., Orihuela, J., Galán, M.R. (2013). Compactness, Optimality, and Risk. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_10

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