Locally nonconical unit balls in Orlicz spaces

The aim of this paper is to investigate the local nonconicality of unit ball in Orlicz spaces, endowed with the Luxemburg norm. A closed convex set Q in a locally convex topological Hausdorff space X is called locally nonconical {LNC), if for every x, y 6 Q there exists an open neighbourhood U of x such that (U Q) + (y - x)/2 C Q- The following theorem is established: An Orlicz space Lv{pi) has an LNC unit ball if and only if either L[fi](u) is finite dimensional or the measure fi is atomic with a positive greatest lower bound and tp satisfies the condition Aj!(/i) and is strictly convex on the interval [0,b], or c[fi] = +oo and [fi] satisfies the condition and is strictly convex on K. A similar result is obtained for the space E[fi](u).