  We derive that for a separable proximinal subspace $Y$ of $X$, $Y$ is
strongly proximinal (strongly ball proximinal) if and only if for $1\leq p<
\infty$, $L_p(I,Y)$ 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 for 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 $1\leq p\leq\infty$. Our observations also
include the fact 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 \cite{LZ}. Several examples are given
having this property, viz. any $U$-subspace of a Banach space, closed unit ball
$B_X$ of a space with $3.2.I.P$, 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.
