  We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional
in $R^N$, and if K is a smooth enough cone, then (modulo constants) u is a
homogenous function of degree 1/2. We deduce some applications in $R^3$ as for
instance that an angular sector cannot be the singular set of a global
minimizer, that if $K$ is a half-plane then $u$ is the corresponding cracktip
function of two variables, or that if K is a cone that meets $S^2$ with an
union of $C^1$ curvilinear convex polygones, then it is a $P$, $Y$ or $T$.
