Best approximation and intersections of balls

Abstract

For a linear subspace G of the normed linear space E and xεE, let P_G(x) be the set of all best approximations of x out of G. Observing that for each x,yεE we always have dist (y,P_G(x))≥‖x−y‖-dist(x,G), we study the subspaces G with the property — which we call property (*) — that this inequality is an equality for each xεE with P_G(x)≠φ and each gεG. This property generalizes the notion of semi L-summand studied by A.Lima. For a subspace G with property (*), the one-sided Gateaux differential of the norm at xεE with OεP_G(x), in the direction gεG equals the distance of −g to the cone spanned by P_G(x). Using this result, we obtain a characterization of those xεE with OεP_G(x) in order that the cone spanned by P_G(x) to be norm-dense in G. When G is proximinal, property (*) is equivalent with 1 1/2-ball property studied by D.Yost. We give geometrical characterizations of the subspaces with property (*), as well as with 1 1/2-ball property.