The Critical Space for Orthogonally Invariant Varieties

Let q be a nondegenerate quadratic form on V. Let X subset of V be invariant for the action of a Lie group G contained in SO(V,q). For any f is an element of V consider the function d(f) from X to C defined by d(f)(x) = q(f - x). We show that the critical points of d(f) lie in the subspace orthogonal to g.f, that we call critical space. In particular any closest point to f in X lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.