  We are concerned with the optimal constants: in the Korn inequality under
tangential boundary conditions on bounded sets $\Omega \subset \mathbb{R}^n$,
and in the geometric rigidity estimate on the whole $\mathbb{R}^2$. We prove
that the latter constant equals $\sqrt{2}$, and we discuss the relation of the
former constants with the optimal Korn's constants under Dirichlet boundary
conditions, and in the whole $\mathbb{R}^n$, which are well known to equal
$\sqrt{2}$. We also discuss the attainability of these constants and the
structure of deformations/displacement fields in the optimal sets.
