  Let $X$ be a reflexive Banach space. In this paper we give a necessary and
sufficient condition for an operator $T\in \mathcal{K}(X)$ to have the best
approximation in numerical radius from the convex subset $\mathcal{U} \subset
\mathcal{K}(X),$ where $\mathcal{K}(X)$ denotes the set of all linear, compact
operators from $X$ into $X.$ We will also present an application to minimal
extensions with respect to the numerical radius. In particular some results on
best approximation in norm will be generalized to the case of the numerical
radius.
