Thin-shell concentration for convex measures (Q2921800)

Let \(X\in\mathbb{R}^n\) be an isotropic random vector and let \(\varepsilon(X)\) be the smallest number \(\varepsilon>0\) such that NEWLINE\[NEWLINE\mathbb{P}\left(\left|\frac{|X|_2}{\sqrt{n}}-1\right|\geq\varepsilon \right)\leq\varepsilon.NEWLINE\]NEWLINENEWLINEIf \(\varepsilon(X)=o(1)\) with respect to the dimension \(n\), it is said that \(X\) is concentrated in a thin shell (notice that \(\mathbb{E}(|X|_2^2)=n\)).NEWLINENEWLINEThin-shell concentration for log-concave measures has been studied in the literature and thus, in particular, the same phenomenon holds for s-concave measures with \(s\geq0\). The main purpose of the paper under review is to prove thin-shell concentration for \(s\)-concave measures in the case \(s<0\). More precisely, the authors show the following result:NEWLINENEWLINELet \(r>2\). Let \(X\in\mathbb{R}^n\) be a full-dimensional \((-1/r)\)-concave random vector. If \(X\) is isotropic, then for any \(p\) such that \(|p|\leq c\min(r,n^{1/3})\), we have NEWLINE\[NEWLINE\alpha_p(X)\leq\frac{C|p-2|}{r}+\left(\frac{C|p-2|}{n^{1/3}}\right)^{3/5},NEWLINE\]NEWLINE where \(C\) and \(c\) are universal constants.NEWLINENEWLINEIn the general case (when \(X\) is not isotropic), let \(A\) be an affine transformation such that \(AX\) is full-dimensional and isotropic. Then, for any \(p\in\mathbb{R}\) such that \(|p|\leq c\min\bigl(r,\frac{n^{1/3}}{\|A\|^{2/3} \|A^{-1}\|^{2/3}}\bigr)\), we have NEWLINE\[NEWLINE\alpha_p(X)\leq\frac{C|p-2|}{r}+\left(\frac{C|p-2|(\|A\|\|A^{-1}\|)^{2/3}}{n^{1/3}}\right)^{3/5},NEWLINE\]NEWLINE where \(C\) and \(c\) are universal constants.NEWLINENEWLINEIndeed, since it can be checked that \(\varepsilon(X)\) is \(o(1)\) if and only if \(\alpha_p(X)\) is \(o(1)\), for \(p>2\) and \(X\) isotropic so that \(|X|_2\) has a finite moment of order \(p\), then the above theorem ensures that any isotropic \((-1/r)\)-concave random vector exhibits thin-shell concentration if \(r\to\infty\) with the dimension \(n\). Hence, almost all of its one-dimensional marginals satisfy a Berry-Esseen theorem.NEWLINENEWLINEThe authors also show sharp reverse Hölder inequalities for \(s\)-concave measures.