On fully rotundity properties and approximative compactness in some Banach sequence spaces

A new notion called compactly fully k-rotundity is introduced in the class of real Banach spaces and it is proved that a real Banach space X is fully k-rotund if and only if it is rotund and compactly fully k-rotund. Some other general results concerning compactly fully k-rotundity, rotundity, fully k-rotundity, compactly locally fully k-routindity are presented. Next, criteria for these properties as well as for approximative compactness in the class of Musielak-Orlicz sequence spaces are obtained.