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, .
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?