Some Geometric Properties of the Cesàro Function Spaces

Some geometric properties of the Ces{\`a}ro function spaces $C_{p,w}$, $1\leqslant p<\infty$, induced by an arbitrary positive weight function $w$ on an interval $(0,l)$ where $0 < l \leqslant\infty$ are studied in this paper. It is shown that all non-empty relatively weakly open sets in the unit ball of $C_{p,w}$ have diameter $2$. Also $C_{p,w}$, $1<p<\infty$ is strictly convex but no point of its unit ball is strongly extreme. Moreover, some connections between uniformly non-square points and various geometric properties in general Banach spaces are presented.