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

A sufficient condition for a Banach space X is given so that every weakly c ompact Chebyshev subset of X is convex. For this purpose a class broader th an that of smooth Banach spaces is defined. In this way a former result of A. Brondsted and A. L. Brown is partially extended in every finite dimensio nal normed linear space and a known result in Hilbert spaces is proved in a different way.