Let q be a nondegenerate quadratic form on V. Let X⊂ V be invariant for the action of a Lie group G contained in SO(V,q). For any f∈ 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

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... , Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.