On Uniform Rotundity in Every Direction in Calderón-Lozanovskii Sequence Spaces

We find a criterion for uniform rotundity in every direction (URED) of Calderón-Lozanovskii 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 Orlicz-Lorentz sequence spaces, which has not been proved until now.