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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)$
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Joram Lindenstrauss, David Preiss, and Jaroslav Tier

Print publication date: 2012

Print ISBN-13: 9780691153551

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153551.001.0001

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Variational Principles

Variational Principles

(p.120) Chapter Seven Variational Principles
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Joram Lindenstrauss

David Preiss

Tiˇser Jaroslav

Princeton University Press

This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where certain functions that change during the process have values close to their infima. Like many other recursive constructions, the choice has a natural description using the language of infinite two-player games with perfect information. The chapter first considers the perturbation game used in Theorem 7.2.1 to formulate an abstract version of the variational principle before showing how to specialize it to more standard formulations. It then examines the bimetric variant of the smooth variational principle, along with the perturbation functions that are relatively simple. It concludes with an assessment of cases when completeness and lower semicontinuity hold only in a bimetric sense.

Keywords:   variational principle, two-player game, perturbation game, perturbation function, completeness, lower semicontinuity

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