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.
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It is proved that the Köthe–Bochner function space E(X) has property β if and only if X is uniformly convex and E has property β. In particular, property β does not lift from X to E(X) in contrast to the case of Köthe–Bochner sequence spaces.
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We study a geometric property in Köthe spaces which is called orthogonal uniform convexity (UC┴). It was introduced in [19]. We prove that the class of Köthe spaces with property (UC┴) is a proper subset of the class of uniformly monotone and P-convex Köthe spaces. Next we consider connections between (UC┴) and property (β) of Rolewicz. We shown that the implication (UC┴) → (β) is true in any Köthe sequence space. Moreover, we find criteria for Orlicz function (sequence) spaces to be orthogonally uniformly convex. As a corollary we get that there holds (UC) → (UC┴) → (β) in any Köthe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) → (β) → (UC┴) hold in any Köthe function space and the second one cannot be reversed.
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