Uniform Kadec-Klee Lorentz spaces L(w,1) and uniformly concave functions

We consider the notion of a uniformly concave function, using it to characterize those Lorentz spaces L(w,1) that have the weak-star uniform Kadec-Klee property as precisely those for which the antiderivative phi of w is uniformly concave; building on recent work of Dilworth and Hsu. We also derive a quite general sufficient condition for a twice-differentiable phi to be uniformly concave; and explore the extent to which this condition is necessary.