Regular cones in Hilbert space (Q1358094)

Let \(H\) be a real Hilbert space of dimension exceeding 1, \(a\) an element of \(H\) with unit norm and \(0<\alpha\leq 1\). The cone \(\{x\in H:(a, x)\geq \alpha|x|\}\) is shown to be regular if and only if \(\alpha=1/\sqrt 2\). The author then goes on to prove that a strictly convex Banach space \(X\) is a Hilbert space if and only if, given \(a, b\in x\) with \(|a-b|=|a+b|\), there is an almost lattical cone containing the elements. (A regular cone \(K\) is almost lattical if for each \(x\in X\), there are \(u, v\in K\) with \(x= u-v\) and \(|u-\lambda v|=|u+\lambda v|\) for each real \(\lambda\)).