On The Fourier Mean Bodies of a Convex Body
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917226974543872 |
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| author | Langharst, Dylan Manui, Auttawich Zvavitch, Artem |
| author_facet | Langharst, Dylan Manui, Auttawich Zvavitch, Artem |
| contents | In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K$ in $\mathbb{R}^n$ for $p>-1$, which now play an important role in geometric tomography. In this work, we study the Fourier transforms of the radial functions of $R_p K$. We introduce a new family of star-shaped sets $F_p K$, which we call the Fourier $p$th mean bodies of $K$. We are then interested in the convexity and the relevant affine-isoperimetric inequalities for $F_p K$, as well as connections of $F_p K$ with other classical objects in geometric tomography such as centroid bodies, intersection bodies, and mean zonoids. We also show that the bodies $F_p K$ are, for $p\in (0,1]$, close to ellipsoids in the sense of Hensley's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20117 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On The Fourier Mean Bodies of a Convex Body Langharst, Dylan Manui, Auttawich Zvavitch, Artem Metric Geometry Functional Analysis Probability 52A40, 52A30, Secondary: 28A75, 42B10 In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K$ in $\mathbb{R}^n$ for $p>-1$, which now play an important role in geometric tomography. In this work, we study the Fourier transforms of the radial functions of $R_p K$. We introduce a new family of star-shaped sets $F_p K$, which we call the Fourier $p$th mean bodies of $K$. We are then interested in the convexity and the relevant affine-isoperimetric inequalities for $F_p K$, as well as connections of $F_p K$ with other classical objects in geometric tomography such as centroid bodies, intersection bodies, and mean zonoids. We also show that the bodies $F_p K$ are, for $p\in (0,1]$, close to ellipsoids in the sense of Hensley's theorem. |
| title | On The Fourier Mean Bodies of a Convex Body |
| topic | Metric Geometry Functional Analysis Probability 52A40, 52A30, Secondary: 28A75, 42B10 |
| url | https://arxiv.org/abs/2601.20117 |