Orlicz affine isoperimetric inequalities for star bodies
Isoperimetric inequality is one of the most important and beautiful geometric inequalities and has a profound impact on geometry and analysis. It is well known that the classical isoperimetric inequality is equivalent to the Sobolev inequality. Unlike the classical isoperimetric inequality, affine isoperimetric inequalities are inequalities between a pair of geometric functionals whose ratio is invariant under affine transformations and have received widespread attention. An interesting inequality in this field that connects the volume of a convex body with that of its polar projection body is the Petty projection inequality. The outstanding work of \textit{G. Zhang} [J. Differ. Geom. 53, No. 1, 183--202 (1999; Zbl 1040.53089)] showed that there exists a Sobolev version inequality that is equivalent to the generalized Petty projection inequality (now known as affine Sobolev-Zhang inequality). Later, the Petty projection inequality has been extended to the \(L_p\) case in [\textit{E. Lutwak} et al., J. Differ. Geom. 56, No. 1, 111--132 (2000; Zbl 1034.52009)] and to the Orlicz case in [Adv. Math. 223, No. 1, 220--242 (2010; Zbl 1437.52006)] by \textit{E. Lutwak} et al. We note that Lutwak-Yang-Zhang only considered the case of convex bodies, and their main tool is Steiner symmetrization. In the present paper, the authors use Steiner symmetrization to prove that the Orlicz Petty projection inequality holds for star bodies. More precisely, they prove that if \(K\) is a Lipschitz star body in \(\mathbb{R}^n\), then the volume ratio \[ \frac{|\Pi^*_{\phi}K|}{|K|} \] is maximized when \(K\) is an ellipsoid centered at the origin. If \(\phi\) is strictly convex, then the ellipsoids centered at the origin are the only maximizers. Here the polar Orlicz projection body \(\Pi^*_{\phi}K\) is an origin-symmetric convex body in \(\mathbb{R}^n\) whose gauge function is defined by \[ \|x\|_{\Pi^*_{\phi}K}=\inf\left\{\lambda>0:\int_{\partial K}\phi\left(\frac{x\cdot\nu_K(y)}{\lambda y\cdot\nu_K(y)}\right)y\cdot\nu_K(y)d\mathcal{H}^{n-1}(y)\leq n|K|\right\}, \] for \(x\in\mathbb{R}^n\), where \(\phi:\mathbb{R}\to[0,\infty)\) is a convex function such that \(\phi(0)=0\), and \(\nu_K(y)\) is the unit normal of \(K\) at \(y\).
scientific article; zbMATH DE number 7461720
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