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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.03977 |
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| _version_ | 1866909379141304320 |
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| author | Lombardi, Nico Richter, Christian Gómez, Eugenia Saorín |
| author_facet | Lombardi, Nico Richter, Christian Gómez, Eugenia Saorín |
| contents | Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension $3$, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03977 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canal Classes and Cheeger Sets Lombardi, Nico Richter, Christian Gómez, Eugenia Saorín Metric Geometry 52A40, 52A15, 52A20, 52A38 Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension $3$, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio. |
| title | Canal Classes and Cheeger Sets |
| topic | Metric Geometry 52A40, 52A15, 52A20, 52A38 |
| url | https://arxiv.org/abs/2411.03977 |