Geometric properties of Orlicz spaces equipped with $p$-Amemiya norms − results and open questions

The classical Orlicz and Luxemburg norms generated by an Orlicz function $\mathrm{\Phi }$ can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573-585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula $\Vert x{\Vert }_{\mathrm{\Phi },p}=\underset{k>0}{inf}\frac{1}{k}(1+I_{\mathrm{\Phi }}^p(kx){)}^{1/p}$, where $1\le p\le \mathrm{\infty }$ (under the convention: $(1+u^{\mathrm{\infty }}{)}^{1/\mathrm{\infty }}=\underset{p\to \mathrm{\infty }}{lim}(1+u^p{)}^{1/p}=max1,u$ for all $u\ge 0$). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Amemiya type norms is investigated. A few open questions concerning this theory will be stated as well.