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 $x,y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+(y-x)/2\subset Q$. The following theorem is established: An Orlicz space $L^{\phi }(\mu )$ has an $LNC$ unit ball if and only if either $L^{\phi }(\mu )$ is finite dimensional or the measure $\mu$ is atomic with a positive greatest lower bound and $\phi$ satisfies the condition ${\delta }_r^0(\mu )$ and is strictly convex on the interval $[0,b]$, or $c(\phi )=+\mathrm{\infty }$ and $\phi$ satisfies the condition ${\mathrm{\Delta }}_2(\mu )$ and is strictly convex on $\mathbb{R}$. A similar result is obtained for the space $E^{\phi }(\mu )$.