Birkhoff-James orthogonality in certain tensor products of Banach spaces II

In this article, we discuss the relationship between Birkhoff-James orthogonality of elementary tensors in the space $L^p(\mu ){\otimes }^{{\mathrm{\Delta }}_p}X,(1\le p<\mathrm{\infty })$ with the individual elements in their respective spaces, where $X$ is a Banach space whose norm is Fr$\acute{e}chet$ differentiable and ${\mathrm{\Delta }}_p$ is the natural norm induced by $L^p(\mu ,X)$. In order to study the said relationship, we first provide some characterizations of Birkhoff-James orthogonality of elements in the Lebesgue-Bochner space $L^p(\mu ,X)$.