Distance sets of well-distributed planar sets for polygonal norms

Let X be a 2-dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX must be if X contains a well-distributed set whose distance set Delta satisfies the estimate |\Delta\cap[0,N]|<CN^{3/2 -\epsilon}. We also give a necessary and sufficient condition for the existence of a well-distributed set with |\Delta \cap [0,N]| < CN.