Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 423-433
doi:10.1155/JIA.2005.423
On moduli of convexity in Banach spaces
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.
Abstract
Let X be a normed linear space, x∈X 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:X→X/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 dimX≥2, then ϱ(x,ε)=ε2/2.