A parametric smooth variational principle and support properties of convex sets and functions |
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Authors: | Libor Veselý |
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Institution: | aDipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy |
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Abstract: | We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets. |
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Keywords: | Convex set Support point Support functional Smooth variational principle Bishop– Phelps theorem |
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