Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 423-433

On moduli of convexity in Banach spaces

Jiří Reif

Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, 306 14 Plzeň, Czech Republic

Received 15 September 2003

Copyright © 2005 Jiří Reif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let X be a normed linear space, xX an element of norm one, and ε>0 and δ(x,ε) the local modulus of convexity of X. We denote by ϱ(x,ε) the greatest ϱ>0 such that for each closed linear subspace M of X the quotient mapping Q:XX/M maps the open ε-neighbourhood of x in U onto a set containing the open ϱ-neighbourhood of Q(x) in Q(U). It is known that ϱ(x,ε)(2/3)δ(x,ε). We prove that there is no universal constant C such that ϱ(x,ε)Cδ(x,ε), however, such a constant C exists within the class of Hilbert spaces X. If X is a Hilbert space with dimX2, then ϱ(x,ε)=ε2/2.