Orlicz difference bodies
In [J. Differ. Geom. 97, No. 3, 427--476 (2014; Zbl 1303.52002)], \textit{R. Gardner} et al., defined the Orlicz-Minkowski combinations of convex bodies. In the paper under review, the authors define the general Orlicz difference body \(\Delta_\varphi^\tau K\) of a convex body \(K\), whose support function is defined by \[h_{\Delta_\varphi^\tau K}(u)=\inf\{\lambda>0:f_1(\tau)\varphi\left(\frac{h_K(u)}{\lambda}\right)+f_2(\tau)\varphi\left(\frac{h_{-K}(u)}{\lambda}\right)\leq\varphi(1)\}. \] Here, \(f_1(\tau)=\frac{(1+\tau)^2}{2(1+\tau^2)}\), \(f_2(\tau)=\frac{(1-\tau)^2}{2(1+\tau^2)}\) for \(\tau\in[-1,1]\) and \(\varphi:[0,\infty)\to[0,\infty)\) is a convex, strictly increasing function such that \(\varphi(0)=0\), \(\varphi(1)=1\). \par This paper deals with the properties of the Orlicz difference bodies \(\Delta_\varphi^\tau K\), which reduce to the \(L_p\) analogs established in [the last two authors, Proc. Am. Math. Soc. 142, No. 7, 2517--2527 (2014; Zbl 1295.52008)] when \(\varphi(t)=t^p\). Also, this article built the following volume comparisons \[V(\Delta_\varphi^\tau K)\geq V(K),\text{ and }V(\Delta_\varphi^{\tau,*}K)\geq V(K^*). \] Here, \(*\) denotes the polars of convex bodies.
scientific article; zbMATH DE number 7045500
scientific article