  A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is
the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of
$K\subseteq \mathbb{Z}^d$ is the function associating to each $u \in
\integer^d$ the cardinality of $K\cap (K+u)$. Daurat, G\'erard, and Nivat and
independently Gardner, Gronchi, and Zong raised the problem on the
reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial
positive answer to this problem by showing that for $d=2$ and under mild extra
assumptions, $g_K$ determines $K$ up to translations and reflections. As a
complement to the theorem on reconstruction we also extend the known
counterexamples (i.e., planar lattice-convex sets which are not
reconstructible, up to translations and reflections) to an infinite family of
counterexamples.
