On some characterizations of inner product spaces (Q1399400)

Let \((X,\|\cdot\|)\) be a real normed linear space, \(x\in X\) and \(A\subseteq X\) a nonempty bounded subset of \(X\). The number \(r(A)=\inf \{r(x,A): x\in X\}\) is called the Chebyshev radius of \(A\), where \(r(x,A)= \sup\{\|x-y\|: y\in A\}\). For spaces \(X\) of dimension \(\geq 3\) the authors prove that \(X\) is an inner product space if and only if it satisfies the following condition: If \(a_1,a_2,a_3\in X\) with \(\|a_1\|=\|a_2\|= \|a_3\|=1\) and \(0\) is in the convex hull of \(\{a_1,a_2,a_3\}\), then \(r(\{a_1,a_2,a_3\})=1\).