On moduli of convexity in Banach spaces

Abstract

Let be a normed linear space, an element of norm one, and and the local modulus of convexity of. We denote by the greatest such that for each closed linear subspace of the quotient mapping maps the open-neighbourhood of in onto a set containing the open-neighbourhood of in. It is known that. We prove that there is no universal constant such that, however, such a constant exists within the class of Hilbert spaces. If is a Hilbert space with, then.