  We analyse the geometric properties of the high derivatives of the distance
function from a submanifold of the Euclidean space. In particular, we show some
relations with the second fundamental form and its covariant derivatives of
independent interest.
  As an application we prove a conjecture of Ennio De Giorgi on the evolution
of submanifolds of the Euclidean space by the gradient of functionals depending
on the derivatives of the distance function.
