Strengthened inequalities for the mean width and the $\ell$-norm
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917396433862656 |
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| author | Böröczky, Károly J. Fodor, Ferenc Hug, Daniel |
| author_facet | Böröczky, Károly J. Fodor, Ferenc Hug, Daniel |
| contents | Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $\ell$-norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of centered isotropic measures on the unit sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_10706 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Strengthened inequalities for the mean width and the $\ell$-norm Böröczky, Károly J. Fodor, Ferenc Hug, Daniel Metric Geometry Primary 52A40, Secondary 52A38, 52B12, 26D15 Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $\ell$-norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of centered isotropic measures on the unit sphere. |
| title | Strengthened inequalities for the mean width and the $\ell$-norm |
| topic | Metric Geometry Primary 52A40, Secondary 52A38, 52B12, 26D15 |
| url | https://arxiv.org/abs/2001.10706 |