Abstract
We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw *-lower semicontinuous function ϕ defined on aw *-compact convex setC in a dual Banach spaceX * and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw *−H σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets.
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Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.
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Poliquin, R.A., Zizler, V.E. Optimization of convex functions on w*-compact sets. Manuscripta Math 68, 249–270 (1990). https://doi.org/10.1007/BF02568763
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DOI: https://doi.org/10.1007/BF02568763