On Minkowski symmetrizations of $α$-concave functions and related applications
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912390181814272 |
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| author | Hoehner, Steven |
| author_facet | Hoehner, Steven |
| contents | A Minkowski symmetral of an $α$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $α$-concave function, there exists a sequence of Minkowski symmetrizations that hypo-converges to its ``hypo-symmetrization". As an application, it is shown that the hypo-symmetrization of a log-concave function $f$ is always harder to approximate than $f$ is by ``inner log-linearizations" with a fixed number of break points. This is a functional analogue of the classical geometric result which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices. Finally, a general extremal property of the hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12619 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Minkowski symmetrizations of $α$-concave functions and related applications Hoehner, Steven Functional Analysis Metric Geometry 52A41 (Primary) 39B62, 52A40 (Secondary) A Minkowski symmetral of an $α$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $α$-concave function, there exists a sequence of Minkowski symmetrizations that hypo-converges to its ``hypo-symmetrization". As an application, it is shown that the hypo-symmetrization of a log-concave function $f$ is always harder to approximate than $f$ is by ``inner log-linearizations" with a fixed number of break points. This is a functional analogue of the classical geometric result which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices. Finally, a general extremal property of the hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases. |
| title | On Minkowski symmetrizations of $α$-concave functions and related applications |
| topic | Functional Analysis Metric Geometry 52A41 (Primary) 39B62, 52A40 (Secondary) |
| url | https://arxiv.org/abs/2301.12619 |