On Mahler's conjecture for even s-concave functions in dimensions 1 and 2
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909659463417856 |
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| author | Fradelizi, Matthieu Nakhle, Elie |
| author_facet | Fradelizi, Matthieu Nakhle, Elie |
| contents | In this paper, we establish different sharp forms of Mahler's conjecture for $s$-concave even functions in dimensions $n$, for $n=1$ and $2$, for $s>-1/n$, thus generalizing our previous results in \cite{FN} on log-concave even functions in dimension 2, which corresponds to the case $s=0$. The functional volume product of an even $s$-concave function $g$ is \[ \int_{\mathbb{R}^{n}}g(x)dx\int_{\mathbb{R}^{n}}\mathcal{L}_{s}g(y)dy, \] where $\mathcal{L}_{s}g$ is the $s$-polar function associated to $g$. The analogue of Mahler's conjecture for even $s$-concave functions postulates that this quantity is minimized for the indicatrix of a cube for any $s>-1/n$. In dimension $n=1$, we prove this conjecture for all $s\in(-1,0)$ (the case $s\ge0$ was established by the first author and Mathieu Meyer in \cite[page 17]{FM10}). In dimension $n=2$, we only consider the case $1/s\in\mathbb{Z}$: for $s>0$, we establish Mahler's conjecture for general $s$-concave even functions; for $s<0$, the situation is more involved, we only prove a sharp inequality for $s$-concave functions $g$ such that $g^s$ admits an asymptote in every direction. Notice that this set of functions is quite natural to consider, when $s<0$, since it is the largest subset of $s$-concave functions stable by $s$-duality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12372 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Mahler's conjecture for even s-concave functions in dimensions 1 and 2 Fradelizi, Matthieu Nakhle, Elie Functional Analysis Metric Geometry In this paper, we establish different sharp forms of Mahler's conjecture for $s$-concave even functions in dimensions $n$, for $n=1$ and $2$, for $s>-1/n$, thus generalizing our previous results in \cite{FN} on log-concave even functions in dimension 2, which corresponds to the case $s=0$. The functional volume product of an even $s$-concave function $g$ is \[ \int_{\mathbb{R}^{n}}g(x)dx\int_{\mathbb{R}^{n}}\mathcal{L}_{s}g(y)dy, \] where $\mathcal{L}_{s}g$ is the $s$-polar function associated to $g$. The analogue of Mahler's conjecture for even $s$-concave functions postulates that this quantity is minimized for the indicatrix of a cube for any $s>-1/n$. In dimension $n=1$, we prove this conjecture for all $s\in(-1,0)$ (the case $s\ge0$ was established by the first author and Mathieu Meyer in \cite[page 17]{FM10}). In dimension $n=2$, we only consider the case $1/s\in\mathbb{Z}$: for $s>0$, we establish Mahler's conjecture for general $s$-concave even functions; for $s<0$, the situation is more involved, we only prove a sharp inequality for $s$-concave functions $g$ such that $g^s$ admits an asymptote in every direction. Notice that this set of functions is quite natural to consider, when $s<0$, since it is the largest subset of $s$-concave functions stable by $s$-duality. |
| title | On Mahler's conjecture for even s-concave functions in dimensions 1 and 2 |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2412.12372 |