Let $X$ be a Banach space, $u\in X^{**}$ and $K, Z$ two subsets of $X^{**}$.
Denote by $d(u,Z)$ and $d(K,Z)$ the distances to $Z$ from the point $u$ and from
the subset $K$ respectively. The Krein-Šmulian Theorem asserts
that the closed convex hull of a weakly compact subset of a Banach space is
weakly compact; in other words, every w$^*$-compact subset $K\subset X^{**}$
such that $d(K,X)=0$ satisfies $d(\overline{\text{co}}^{w^*}(K),X)=0$. We extend
this result in the following way: if $Z\subset X$ is a closed subspace of $X$
and $K\subset X^{**}$ is a w$^*-$compact subset of $X^{**}$, then $$
d(\overline{\text{co}}^{w^*}(K),Z)\leq 5 d(K,Z). $$ Moreover, if $Z\cap K$ is
w$^*$-dense in $K$, then $d(\overline{\text{co}}^{w^*}(K),Z)\leq 2 d(K,Z)$.
However, the equality $d(K,X)=d(\overline{\text{co}}^{w^*}(K),X)$ holds in many
cases, for instance, if $\ell_1\not\subseteq X^*$, if $X$ has w$^*$-angelic dual
unit ball (for example, if $X$ is WCG or WLD), if $X=\ell_1(I)$, if $K$ is
fragmented by the norm of $X^{**}$, etc. We also construct under $CH$ a
w$^*$-compact subset $K\subset B(X^{**})$ such that $K\cap X$ is w$^*$-dense in
$K$, $d(K,X)=\frac 12$ and $d(\overline{\text{co}}^{w^*}(K),X)=1$.