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Journal of Convex Analysis, Vol. 6, No. 1, pp. 91-113 (1999)
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Some Geometric Properties in Orlicz Sequence Spaces equipped with Orlicz Norm

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Yunan Cui and Henryk Hudzik and Marian Nowak and Ryszard Pluciennik

Department of Mathematics, Harbin University of Science and Technology, Xuefu Road 52, 150080 Harbin, China, yunancui@public.hr.hl.cn and Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Ul. Matejki 48/49, 60-769 Poznan, Poland, hudzik@amu.edu.pl and Institute of Mathematics, T. Kotarbinski Pedagogical University, Pl. Slowianski 9, 65-069 Zielona Gora, Poland, nowakmar@omega.im.wsp.zgora.pl and Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland, rplucien@math.put.poznan.pl

**Abstract:** It is proved that for any reflexive Banach space $X$, both $X$ and $X^{*}$ are ** CLUR** if and only if both $X$ and $X^{*}$ have property ** H**. Criteria for rotundity, local uniform rotundity, compact local uniform rotundity and property ** H** in Orlicz sequence spaces equipped with the Orlicz norm are given. Criteria for property ** H**, rotundity and ** LUR** were already known in the literature only for finitely valued Orlicz functions which vanish only at zero and are $N$-functions (i.e. they satisfy conditions $(0_1)$ and $(\infty_1)$. All our criteria except Corollary 2.15 are given for arbitrary Orlicz functions. Criteria for smoothness of $l_\Phi^0$ in Corollary 2.15 are given for any finitely valued Orlicz function satisfying condition $(\infty_1)$, extending the respective result of [2] proved only for Orlicz functions vanishing only at zero.

**Keywords:** Orlicz sequence space, rotundity, local uniform rotundity, compact local uniform rotundity, property ** H**, smoothness, copy of $l_\infty$

**Classification (MSC2000):** 46E30, 46E40, 46B20

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