Linear bijections which preserve the diameter of vector-valued maps

We study diameter preserving linear bijections from C(X,V) onto C(Y,Z), where X,Y are compact Hausdorff spaces and V, Z are Banach spaces. In particular, assuming that Z is rotund and the extreme points of BV∗ satisfy a certain geometric condition, we prove that there exists a diameter preserving linear bijection from C(X,V) onto C(Y,Z) if and only if X is homeomorphic to Y and Z is linearly isometric to V. We also consider the case when X and Y are locally compact, noncompact spaces.