The convex hull property and topology of hypersurfaces with nonnegative curvature |
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Authors: | Stephanie Alexander Mohammad Ghomi |
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Affiliation: | a Department of Mathematics, University of Illinois, Urbana, IL 61801, USA b Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA |
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Abstract: | We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure. |
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Keywords: | primary 53A07 53C45 secondary 53C21 53C23 |
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