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Dualistic Riemannian Manifold Structure Induced from Convex Functions

  • Jun Zhang
  • Hiroshi Matsuzoe
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 17)

Abstract

Convex analysis has wide applications in science and engineering, such as mechanics, optimization and control, theoretical statistics, mathematical economics and game theory, and so on. It offers an analytic framework to treat systems and phenomena that depart from linearity, based on an elegant mathematical characterization of the notion of “duality” (Rockafellar, 1970, 1974, Ekeland and Temam, 1976). Recent work of David Gao (2000) further provided a comprehensive and unified treatment of duality principles in convex and nonconvex systems, greatly enriching the theoretical foundation and scope of applications.

Key words

Legendre–Fenchel duality biorthogonal coordinates Riemannian metric conjugate connections equiaffine geometry parallel volume form affine immersion Hessian geometry 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of MichiganAnn ArborU.S.A
  2. 2.Department of Computer Science and EngineeringNagoya Institute of TechnologyNagoyaJapan

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