Mean curvature and closed geodesics in convex hypersurfaces

James Dibble; Joseph Ansel Hoisington

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arXiv · 2606.04112

Mean curvature and closed geodesics in convex hypersurfaces

James Dibble·Joseph Ansel Hoisington

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Abstract

We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of \'{A}lvarez Paiva for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible.

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James Dibble, Joseph Ansel Hoisington. 2026-06-02. Mean curvature and closed geodesics in convex hypersurfaces. https://arxiv.org/abs/2606.04112

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