Flat spots and super-reflexivity in normed spaces

AbstractLet X be a normed space, x∈X||x||=1. Then m(x) is the infimum of the lengths of all curves in the unit sphere of X, connecting x and −x, and m(X):=inf {m(x):||x|| = 1}. If m(x)=2, then x; is a flat spot.It is proved here, that, if m(X)=2 (equivalently, if the completion of X is not super-reflexive), then, for every x0,# 0, X admits an equivalent norm for which xo is a flat spot