Let \(X\) be a Banach space and \(E\) be a closed bounded subset of \(X\). For \(x \in X\) we set \(D(x,E) = \sup\{\| x − e \|: e \in E\}\). The set \(E\) is called remotal in \(X\) if for any \(x \in X\), there exists \(e \in E\) such that \(D(x,E) = \| x − e \|\) . It is the object of this paper to give new results on remotal sets in \(L^p(I,X)\), and to simplify the proofs of some results in [5].