An asymmetric Orlicz centroid inequality for probability measures
Let \(\operatorname{Conv}(\mathbb{R})\) be the class of convex functions \(\phi: \mathbb{R}\to [0, \infty)\), \(\phi(0)=0\) and such that \(\phi\) is either strictly decreasing on \((-\infty, 0]\) or \(\phi\) is strictly increasing on \([0, \infty)\). Denote by \(D_n\) the Euclidean ball of volume one. Lebesgue measure restricted to \(D_n\) is \(\lambda_{D_n}\). Let \({\mathcal{P}}_{[n]}\) be the class of probability measures on \(\mathbb{R}^n\) that are absolutely continuous with respect to Lebesgue measure. For a convex body \(K\) (that is a compact convex set in \(\mathbb{R}^n\) with non-empty interior) denote by \(h(K, y) =\max\{x\cdot y: x\in K\}\), for \(y\in K\), the support function of \(K\). In [\textit{E. Lutwak} et al., J. Differ. Geom. 84, No. 2, 365--387 (2010; Zbl 1206.49050)] has been proved the following Orlicz centroid inequality for convex bodies: Let \(\phi\in \operatorname{Conv}(\mathbb{R})\) and \(K\) be a convex body in \(\mathbb{R}^n\) that contains the origin in its interior with \(vol_n(K)=1\). Define the Orlicz centroid body \(\Gamma_{\phi}K\) to be the convex body whose support function at \(y\in \mathbb{R}^n\) is given by \(h(\Gamma_{\phi}K, y) = \inf\left\{\lambda>0: \int_K\phi\left(x\cdot y/\lambda\right)\, dx\leq 1\right\}\). Then, \(vol_n(\Gamma_{\phi}K) \geq vol_n(\Gamma_{\phi}D_n)\), equality holds if and only if \(K\) is an ellipsoid centered at the origin. Haberl and Schuster's asymmetric \(L_p\) centroid inequality in [\textit{C. Haberl} and \textit{F. E. Schuster}, J. Differ. Geom. 83, No. 1, 1--26 (2009; Zbl 1185.52005)] is the case \(\phi(t) =(|t| +\alpha t)^p\), for \(-1\leq \alpha \leq 1\) of the inequality. By using probability arguments, \textit{G. Paouris} and \textit{P. Pivovarov} in [Adv. Math. 230, No. 3, 1402--1422 (2012; Zbl 1260.52006)] obtained the symmetric Orlicz centroid inequality for probability measures: Let \(\phi\in \operatorname{Conv}[0, \infty)\) and \(\mu\in {\mathcal{P}}_{[n]}\). Define the symmetric Orlicz centroid body \(\Gamma_{\phi}(\mu)\) of \(\mu\) corresponding to \(\phi\) by its support function \(h(\Gamma_{\phi}(\mu), y) = \inf\left\{\lambda>0: \int_{\mathbb{R}^n}\phi\left(|x\cdot y|/\lambda\right)\, d\mu\leq 1\right\}\), and \(f\) denotes the density of \(\mu\). If \(\|f\|_{\infty}\leq 1\), then \(vol_n(\Gamma_{\phi}(\mu)) \geq vol_n(\Gamma_{\phi}(\lambda_{D_n}))\). The main purpose of this paper is to exhibit the asymmetric version of the above result. The rest of the paper is organized as follows: In Section 2, some basic notation and preliminaries are provided. In Section 3, the proof of the mail result (asymmetric Orlicz centroid inequality for probability measures) is obtained. In Section 4, the asymmetric \(L_p\) centroid inequality for compact sets is given, which extends the result from [Haberl and Schuster, loc. cit.].
scientific article; zbMATH DE number 6376114
scientific article