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 Φ 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 ∥x∥Φ,p=infk>011/k(1+IpΦ(kx))1/p, where 1≤p≤∞ (under the convention: (1+u∞)1/∞=limp→∞(1+up)1/p=max1,u for all u≥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.