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Part of the book series: Mathematics and Its Applications ((MAIA,volume 297))

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Abstract

On noncompact (paracompact) ndimensional manifold M one can formulate the following questions:

  1. 1)

    Are there any complete Riemannian metric g on M and a function f: M → ℝ such that f is convex relative to g ?

  2. 2)

    Being given a complete Riemannian metric g on M, is there any function f: M → ℝ which is convex with respect to g ?

  3. 3)

    Being given a function f: M → ℝ which is convex with respect to the complete Riemannian metric g on M, what kind of changes of the metric g preserve the completeness or the convexity of f ?

  4. 4)

    Being given a C2 function f:M → ℝ without critical points which might be maximum points of f or inflexion points of the graph of f, is there any complete Riemannian metric g on M relative to which f becomes a convex function ?

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© 1994 Springer Science+Business Media Dordrecht

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Udrişte, C. (1994). Geometric Examples of Convex Functions. In: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, vol 297. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8390-9_4

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  • DOI: https://doi.org/10.1007/978-94-015-8390-9_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4440-2

  • Online ISBN: 978-94-015-8390-9

  • eBook Packages: Springer Book Archive

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