Criteria for compactly locally uniformly rotund points in Musielak-Orlicz spaces equipped with the Luxemburg and the Orlicz-Amemiya norms are given. Next, criteria for compact local uniform rotundity and local uniform rotundity of the spaces for both norms are deduced. These properties are important because, for any Banach space X, both of them imply the Kadec-Klee property and this property, together with reflexivity, is equivalent to approximative compactness of X. Approximative compactness of X gives that any nonempty convex and closed set in X is proximinal in X and the projection PA( . ) from X to A is a continuous operator [see I. Singer: The Theory of Best Approximation and Functional Analysis. Springer-Verlag 1970].
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Henryk Hudzik, Wojciech Kowalewski, On Some Local Geometric Properties in Musielak-Orlicz Function Spaces. Z. Anal. Anwend. 23 (2004), no. 4, pp. 683–712DOI 10.4171/ZAA/1216