  In this paper we study the space $\mathcal{M}$ of all nonempty compact metric
spaces considered up to isometry, equipped with the Gromov--Hausdorff distance.
We show that each ball in $\mathcal{M}$ with center at the one-point space is
convex in the weak sense, i.e., every two points of such a ball can be joined
by a shortest curve that belongs to this ball, however, such a ball is not
convex in the strong sense: it is not true that every shortest curve joining
the points of the ball belongs to this ball. We also show that a ball of
sufficiently small radius with center at a space of general position is convex
in the weak sense.
