Approximate Birkhoff-James orthogonality preserver on Lebesgue-Bochner spaces

In this article, we examine an approximate version of Koldobsky-Blanco-Turnšek theorem (namely, Property P) in the space of vector-valued integrable functions. More precisely, we prove that the Lebesgue-Bochner spaces $L^p(\mu ,X),(1\le p<\mathrm{\infty })$, do not have Property P under certain conditions on $\mu$ and the Banach space $X$.