On some local geometric properties in Musielak-Orlicz function spaces (Q1770171)

A point \(x\) in the unit sphere \(S(X)\) of a Banach space \(X\) is a point of compact local uniform rotundity (of the unit ball of \(X\)) if the set \(\{x_n\}\) is a relatively compact set in \(S(X)\) whenever \((x_n)\) is a sequence in \(S(X)\) with \(\| x_n + x \| \rightarrow 2\). In this article, the authors characterize the points of compact local uniform rotundity of the unit ball of Musielak-Orlicz spaces in terms of the Orlicz function generating the space and its conjugate function. The results assume mild restrictions on the Orlicz functions; the measure space is assumed to be nonatomic, complete, and \(\sigma\)-finite; and characterizations are given when the Musielak-Orlicz space is endowed with either the Luxemburg norm or the Orlicz-Amemiya norm. As a corollary, the authors note that, endowed with either norm, a Musielak-Orlicz space is locally uniformly rotund exactly when it is compactly locally uniformly rotund.