# Convexity of the modulus of convexity

The book "Geometric Properties of Banach Spaces and Nonlinear Iterations" by Charles Chidume (https://link.springer.com/book/10.1007/978-1-84882-190-3) says (page 7)

"We now present the following known and interesting theorem. A proof can be found in Lindenstrauss and Tzafriri, [312].

Theorem 1.13. The modulus of convexity of a normed space $$X$$, $$\delta_X$$, is a convex and continuous function."

However Lindenstrauss and Tzafriri's book "Classical Banach Spaces, vol. 2" (page 67) says "The modulus of convexity of a Banach space need not be itself a convex function."

So it seems that Chidume's book is incorrect, but it seems like a very major error. Can anyone sheed some light on the situation?

• Does the book by Lindenstrauss and Tzafriri has a counterexample? Have you tried to compute the modulus of convexity for some simple norms? – gerw Jun 11 at 13:39