On the convexity of the weakly compact Chebyshev sets in Banach spaces

A sufficient condition for a Banach spaceX is given so that every weakly compact Chebyshev subset ofX is convex. For this purpose a class broader than that of smooth Banach spaces is defined. In this way a former result of A. Brøndsted and A. L. Brown is partially extended in every finite dimensional normed linear space and a known result in Hilbert spaces is proved in a different way.