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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?

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  • $\begingroup$ Does the book by Lindenstrauss and Tzafriri has a counterexample? Have you tried to compute the modulus of convexity for some simple norms? $\endgroup$ – gerw Jun 11 at 13:39
  • $\begingroup$ See also en.wikipedia.org/wiki/… $\endgroup$ – gerw Jun 11 at 13:41
  • $\begingroup$ Wikipedia just references Lindenstrauss and Tzafriri's book. The book says that counterexamples are in Liokoumovich 1973 article "The existence of B-spaces with non-convex modulus of convexity" which is in Russian. $\endgroup$ – Quarto Bendir Jun 11 at 15:25
  • $\begingroup$ Basically I am looking for an expert to clarify whether Chidume's statement is correct in some special setting, or whether it is just completely wrong $\endgroup$ – Quarto Bendir Jun 11 at 15:26

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