A note on measures of nonconvexity

We define the Hausdorff measure of nonconvexity β(C)β(C) of a nonempty bounded subset CC of a Banach space XX as the Hausdorff distance of CC to the family of all the nonempty convex bounded subsets of XX. We compare the measure ββ with the Eisenfeld–Lakshmikantham measure of nonconvexity αα and prove that the two measures are equivalent (β≤α≤2β)(β≤α≤2β), but in general they are different.