Convex domains of Finsler and Riemannian manifolds |
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Authors: | Rossella Bartolo Erasmo Caponio Anna Valeria Germinario Miguel S��nchez |
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Institution: | 1. Dipartimento di Matematica, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy 2. Dipartimento di Matematica, Universit?? degli Studi di Bari, Via Orabona 4, 70125, Bari, Italy 3. Departamento de Geometr??a y Topolog??a, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071, Granada, Spain
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Abstract: | A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop??s one in his classical result (Bishop, Indiana Univ Math J 24:169?C172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface. |
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