On Moduli of Convexity in Banach Spaces

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.