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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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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