Some geometric properties of spaces of vector-valued integrable functions

We identify the smooth points of $L^1(\mu ,X)$, and provide some necessary and sufficient conditions for left and right symmetry of points with respect to Birkhoff-James orthogonality in $L^p(\mu ,X),1\le p<\mathrm{\infty }$, where $\mu$ is any complete positive measure and $X$ is a Banach space with some suitable properties.