
Journal of Convex Analysis 14 (2007), No. 2, 395412 Copyright Heldermann Verlag 2007 Local Uniform Rotundity in CalderónLozanovskii Spaces Pawel Foralewski Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61614 Poznan, Poland katon@amu.edu.pl Pawel Kolwicz Institute of Mathematics, Faculty of Electricity, University of Technology, Piotrowo 3a, 60965 Poznan, Poland kolwicz@math.put.poznan.pl We find criteria for local uniform rotundity of CalderonLozanovskii 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)] and generalizing several theorems, which give only the sufficient (or necessity) conditions [see H. Hudzik, A. Kaminska and M. Mastylo, "Monotonicity and rotundity properties in Banach lattices", Rocky Mountain J. Math. 30(3) (2000) 933949; cf. also J. Cerda, H. Hudzik and M. Mastylo, "On the geometry of some CalderonLozanovskii interpolation spaces", Indagationes Math. N.S. 6(1) (1995) 3549]. In particular we obtain the respective criteria for OrliczLorentz spaces which has been proved directly in papers of H. Hudzik, A. Kaminska and M. Mastylo ["On geometric properties of OrliczLorentz spaces", Canadian Math. Bull. 40(3) (1997) 316329] and J. Cerda, H. Hudzik, A. Kaminska and M. Mastylo ["Geometric properties of symmetric spaces with applications to OrliczLorentz spaces", Positivity 2 (1998) 311337]. Keywords: Koethe space, CalderonLozanovskii space , OrliczLorentz space, local uniform rotundity, monotonicity properties. MSC: 46B20, 46E30, 46B04, 46B42, 46A45 [ Fulltextpdf (183 KB)] for subscribers only. 