Convexity of Radial Mean Bodies via an extension of Ball's Bodies
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908988823568384 |
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| author | Langharst, Dylan |
| author_facet | Langharst, Dylan |
| contents | In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable log-concave function which attains its maximum at the origin, then \[ x\mapsto \left(\frac{p}{g(o)}\int_{0}^{\infty}r^{p-1}(g(rx)-g(o))\mathrm{d}\,r\right)^{-\frac{1}{p}} \] is a positively 1-homogeneous convex function on $\mathbb{R}^n$. Our approach also provides a new proof of the original regime $p> 0$. The argument is based on a reduction to a two-dimensional inequality derived from Prékopa's theorem, which may be of independent interest.
As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative. In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K\subset \mathbb{R}^n$ for $p>-1$. Furthermore, they established that $R_p K$ is convex for $p\geq 0$, but the convexity of $R_p K$ for $p\in (-1,0)$ remained open. We prove that $R_p K$ is convex for all $p>-1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14134 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convexity of Radial Mean Bodies via an extension of Ball's Bodies Langharst, Dylan Metric Geometry Functional Analysis Primary 52A20, Secondary 52A30, 26B25 In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable log-concave function which attains its maximum at the origin, then \[ x\mapsto \left(\frac{p}{g(o)}\int_{0}^{\infty}r^{p-1}(g(rx)-g(o))\mathrm{d}\,r\right)^{-\frac{1}{p}} \] is a positively 1-homogeneous convex function on $\mathbb{R}^n$. Our approach also provides a new proof of the original regime $p> 0$. The argument is based on a reduction to a two-dimensional inequality derived from Prékopa's theorem, which may be of independent interest. As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative. In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K\subset \mathbb{R}^n$ for $p>-1$. Furthermore, they established that $R_p K$ is convex for $p\geq 0$, but the convexity of $R_p K$ for $p\in (-1,0)$ remained open. We prove that $R_p K$ is convex for all $p>-1$. |
| title | Convexity of Radial Mean Bodies via an extension of Ball's Bodies |
| topic | Metric Geometry Functional Analysis Primary 52A20, Secondary 52A30, 26B25 |
| url | https://arxiv.org/abs/2603.14134 |