  Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it
has been recently proved in spherical and the hyperbolic cases, as well, that
Reuleaux triangles have the minimal area among convex domains of constant width
$D$. We prove an essentially optimal stability version of this statement in
each of the three types of surfaces of constant curvature. In addition, we
summarize the fundamental properties of convex bodies of constant width in
spaces of constant curvature, and provide a characterization in the hyperbolic
case in terms of horospheres.
