  A closed, convex set $K$ in $\mathbb{R}^2$ with non-empty interior is called
lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^2$. In this
paper we study the relation between the area and the lattice width of a planar
lattice-free convex set in the general and centrally symmetric case. A
correspondence between lattice width on the one hand and covering minima on the
other, allows us to reformulate our results in terms of covering minima
introduced by Kannan and Lov\'asz. We obtain a sharp upper bound for the area
for any given value of the lattice width. The lattice-free convex sets
satisfying the upper bound are characterized. Lower bounds are studied as well.
Parts of our results are applied in a paper by the authors and Weismantel for
cutting plane generation in mixed integer linear optimization, which was the
original inducement for this paper. We further rectify a result of Kannan and
Lov\'asz with a new proof.
