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In the first part of this paper we study a best approximation of a vector inEuclidean space R^n with respect to a closed semi-algebraic set C and a givensemi-algebraic norm. Assuming that the given norm and its dual norm aredifferentiable we show that a best approximation is unique outside ahypersurface. We then study the case where C is an irreducible variety and theapproximation is with respect to the Euclidean norm. We show that for a generalpoint in x in R^n the number of critical points of the distance function of xto C is bounded above by a degree of a related dominant map. If C induces asmooth projective variety in V_P in P(C^n) then this degree is the top Chernnumber of a corresponding vector bundle on V_P. We then study the problem whena best k(> 1)-border rank approximation of a symmetric tensor is symmetric. Weshow that under certain dimensional conditions there exists an opensemi-algebraic set of symmetric tensors for which a best k-border rank isunique and symmetric.

Best approximation on semi-algebraic sets and k-border rank approximation of symmetric tensors