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The classical Orlicz and Luxemburg norms generated by an Orlicz function $\Phi$ 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 ${\parallel x\parallel }_{\Phi ,p}={inf}_{k>0}\frac{1}{k}{(1+I_{\Phi }^p\left(kx\right))}^{1/p}$, where $1\le p\le \infty$ (under the convention: ${(1+u^{\infty })}^{1/\infty }={lim}_{p\to \infty }{(1+u^p)}^{1/p}=max1,u$ for all $u\ge 0$). Based on this idea, a number of papers have been published in the past few years. In this...

The aim of this paper is to investigate stability of unit ball in Orlicz spaces, endowed with the Luxemburg norm, from the “local” point of view. Firstly, those points of the unit ball are characterized which are stable, i.e., at which the map $z\to \{(x,y):\frac{1}{2}(x+y)=z\}$ is lower-semicontinuous. Then the main theorem is established: An Orlicz space $L^{\varphi }\left(\mu \right)$ has stable unit ball if and only if either $L^{\varphi }\left(\mu \right)$ is finite dimensional or it is isometric to $L^{\infty }\left(\mu \right)$ or $\varphi$ satisfies the condition ${\Delta }_r$ or ${\Delta }_r^0$ (appropriate to the measure $\mu$ and the function...

In the paper we consider a class of Orlicz spaces equipped with the Orlicz norm over a non-negative, complete and sigma-finite measure space (T,Sigma,mu), which covers, among others, Orlicz spaces isomorphic to L-infinite and the interpolation space L1 + L-infinite. We give some necessary conditions for a point x from the unit sphere to be extreme. Applying this characterization, in the case of an atomless measure mu, we find a description of the set of extreme points of L1 + L-infinite which corresponds...

Let $E^{\varphi }\left(\mu \right)$ be the subspace of finite elements of an Orlicz space endowed with the Luxemburg norm. The main theorem says that a compact linear operator $T:E^{\varphi }\left(\mu \right)\to C\left(\Omega \right)$ is extreme if and only if $T^{*}\omega \in ExtB\left({\left(E^{\varphi }\left(\mu \right)\right)}^{*}\right)$ on a dense subset of $\Omega$, where $\Omega$ is a compact Hausdorff topological space and $\langle T^{*}\omega ,x\rangle =\left(Tx\right)\left(\omega \right)$. This is done via the description of the extreme points of the space of continuous functions $C(\Omega ,L^{\varphi }\left(\mu \right))$, $L^{\varphi }\left(\mu \right)$ being an Orlicz space equipped with the Orlicz norm (conjugate to the Luxemburg one). There is also given a theorem on closedness of the set of extreme...