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Invariants in the Riemannian geometry of convex sets

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  • HILDEBRAND, Roland

Abstract

In this contribution we study some aspects of the Riemannian geometry induced on a convex set by a barrier function of the set. Using Noether's theorem, we link the symmetries of the set to invariants of the geodesic flow. This allows to lower the dimension of the differential system defining the geodesics and gives insights in the structure of the geodesic flow, specifically on the configuration of geodesic submanifolds. We use the developed apparatus to completely integrate the geodesicequations for the convex hulls of the sphere, the paraboloid, the hyperboloid and the standard symmetricc ones and to obtain explicit formulae for the geodesics on these sets.

Suggested Citation

  • HILDEBRAND, Roland, 2004. "Invariants in the Riemannian geometry of convex sets," LIDAM Discussion Papers CORE 2004007, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2004007
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp2004.html
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    More about this item

    Keywords

    Riemannian geometry; barrier function; convexity; symmetry;
    All these keywords.

    JEL classification:

    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other

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