Brunn-Minkowski inequalities for sprays on surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909315366912000 |
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| author | Assouline, Rotem |
| author_facet | Assouline, Rotem |
| contents | We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn-Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function $κ$ on the surface, and $κ$ satisfies the inequality $$K + κ^2 - |\nablaκ| \ge 0$$ where $K$ is the Gauss curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_04585 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Brunn-Minkowski inequalities for sprays on surfaces Assouline, Rotem Differential Geometry Metric Geometry We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn-Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function $κ$ on the surface, and $κ$ satisfies the inequality $$K + κ^2 - |\nablaκ| \ge 0$$ where $K$ is the Gauss curvature. |
| title | Brunn-Minkowski inequalities for sprays on surfaces |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2211.04585 |