Let K(X,Y) be the space of compact operators. For a proximinal subspace Z⊂Y, this paper deals with question, when does every Y-valued operator admit Z-valued best approximation? any reflexive Banach X and for L1-predual Y, if Z⊂Y is strongly finite codimension, we show that K(X,Z) under an additional condition on position K(X,Z). When Y c0-direct sum dimensional spaces achieve strong transitivi...

Lasserre [32] proved that for every compact set K⊂Rn and even number d there exists a unique homogeneous polynomial g0 of degree with K⊂G1(g0)={x∈Rn:g0(x)≤1} minimizing |G1(g)| among all such polynomials g fulfilling the condition K⊂G1(g). This result extends notion Löwner ellipsoid, not only from convex bodies to arbitrary sets (which was immediate if d=2 by taking hulls), but also ellipsoids ...