
Journal of Convex Analysis 14 (2007), No. 3, 621645 Copyright Heldermann Verlag 2007 On Uniform Rotundity in Every Direction in CalderónLozanovskii Sequence Spaces Pawel Kolwicz Inst. of Mathematics, Poznán University of Technology, Piotrowo 3A, 60965 Poznán, Poland kolwicz@math.put.poznan.pl Ryszard Pluciennik Inst. of Mathematics, Poznán University of Technology, Piotrowo 3A, 60965 Poznán, Poland rplucien@math.put.poznan.pl We find a criterion for uniform rotundity in every direction (URED) of CalderónLozanovskii sequence spaces solving Problem XII from S. T. Chen, Y. A. Cui, H. Hudzik and T. F. Wang ["On some solved and unsolved problems in geometry of certain classes of Banach function spaces", in: Unsolved Problems on Mathematics for the 21st Century, J. M. Abe & Tanaka (eds.), IOS Press (2001)]. In order to do it, we study properties of the directed modulus of convexity of Banach spaces. Next we introduce and study new notions such as uniform rotundity in every interval (UREI) and uniform monotonicity in every interval (UMEI). They are crucial to get the main criterion, that is a Köthe sequence space is UREI iff it is URED and order continuous. Then we show the important (for further investigations) characterization of the property UREI on the positive cone of Köthe sequence space. Applying that we prove the characterization mentioned at the beginning of the abstract. As a corollary, we obtain the criterion for URED of OrliczLorentz sequence spaces, which has not been proved until now. Keywords: Koethe space, CalderonLozanovskii spaces, OrliczLorentz space, uniform rotundity in every direction. MSC: 46E30, 46B20, 46B42, 46A45 [ Fulltextpdf (224 KB)] for subscribers only. 