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.