Rotundity properties in Banach spaces via sublinear operators

Conditions for extreme points, strongly extreme points, LUR-points, SU-points, points in direction of which the space is uniformly rotund and points with the Kadec–Klee property with respect to the convergence μμ-almost everywhere for the space DE(S)DE(S), generated by a Köthe space E and a sublinear operator S   defined on a linear space XX and with values in L0L0, are given. As corollaries, conditions for rotundity, midpoint local uniform rotundity, local uniform rotundity, uniform rotundity in every direction and the Kadec–Klee property for the convergence locally in measure of these spaces are deduced. Interpretations of the results in cases of Köthe–Bochner spaces, Cesáro–Orlicz spaces and K- interpolation spaces are presented.