Extremal affine surface areas in a functional setting
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916138410049536 |
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| author | Egler, Stephanie Werner, Elisabeth M. |
| author_facet | Egler, Stephanie Werner, Elisabeth M. |
| contents | We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows that the maximal (resp. minimal) inner affine surface area of an $s$-concave function on $\mathbb{R}^n$ equals the maximal (resp. minimal) outer affine surface area of its Legendre polar. We estimate the ``size" of these quantities: up to a constant depending on $n$ and $s$ only, the extremal affine surface areas are proportional to a power of the integral of $f$. This extends results obtained in the setting of convex bodies. We recover and improve those as a corollary to our results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremal affine surface areas in a functional setting Egler, Stephanie Werner, Elisabeth M. Metric Geometry Functional Analysis We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows that the maximal (resp. minimal) inner affine surface area of an $s$-concave function on $\mathbb{R}^n$ equals the maximal (resp. minimal) outer affine surface area of its Legendre polar. We estimate the ``size" of these quantities: up to a constant depending on $n$ and $s$ only, the extremal affine surface areas are proportional to a power of the integral of $f$. This extends results obtained in the setting of convex bodies. We recover and improve those as a corollary to our results. |
| title | Extremal affine surface areas in a functional setting |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2402.16130 |