We derive that a separable proximinal subspace $Y$ of $X$, $Y$ is strongly proximinal (strongly ball proximinal) if and only if $1\leq p<\infty$, $L_p(I,Y)$ is strongly proximinal (strongly ball proximinal) in $L_p(I,X)$. Case for $p=\infty$ follows from stronger assumption on $Y$ in $X$ (uniform proximinality). It is observed that a separable proximinal subspace $Y$ in $X$, $Y$ is ball proximinal in $X$ if and only if $L_p(I,Y)$ is ball proximinal in $L_p(I,X)$ for $1\leq p\leq\infty$; this observation also extends to that for any (strongly) proximinal subspace $Y$ of $X$, if every separable subspace of $Y$ is ball (strongly) proximinal in $X$ then $L_p(I,Y)$ is ball (strongly) proximinal in $L_p(I,X)$ for $1\leq p<\infty$. We introduce the notion of uniform proximinality of a closed convex set in a Banach space, which is wrongly defined in a recent paper by P.-K Lin et al[J. Approx. Theory 183 (2014), 72--81]. Several examples are given having this property, viz. any $U$-subspace of a Banach space has this property. Recall the notion of $3.2.I.P.$ by Joram Lindenstrauss, a Banach space $X$ is said to have $3.2.I.P.$ if any three closed balls which are pairwise intersecting actually intersect in $X$. It is proved the closed unit ball $B_X$ of a space with $3.2.I.P$ and closed unit ball of any M-ideal of a space with $3.2.I.P.$ are uniformly proximinal. A new class of examples are given having this property.