Whitney Regularity of the Image of the Chevalley mapping

A closed set $F$ is Whitney 1-regular if for each compact $K\subset F$, the geodesic distance in $K$ is equivalent to the Euclidean distance. Let $P$ be the Chevalley map defined by an integrity basis of the algebra of polynomials invariant by a reflection group, this note gives the Whitney regularity of the image by $P$ of closed balls centered at the origin of ${\mathbb R}^n$. The proof uses the works of Givental', Kostov and Arnold on the symmetric group. It needs a generalization of a property of the Van der Monde determinants to the Jacobian of the Chevalley mappings.