A characterization of Euclidean norms in finite dimension (Q1813179)

The author proves a characterization of finite-dimensional Hilbert spaces: Let \(E\) be a finite-dimensional real normed space and put \(B=\{T: E\to E\mid\) \(T\) linear, \(\| T\|\leq 1\}\), \(G=\{T\in B\mid\) \(T\) is isometric\(\}\). Then the norm of \(E\) is Euclidean if and only if \(G\) coincides with the extreme point set of \(B\). Then if part of this result is closely related to the notion of convex transitivity (not mentioned here): For each \(x\in E\) with \(\| x\|=1\) put \(G(x)=\) closed convex hull of \(\{Tx\mid\;T\in G\}\). If \(G=\) extreme point set of \(B\), then a straightforward application of the Krejn-Mil'man theorem yields that \(B\) leaves \(G(x)\) invariant. This implies easily \(G(x)=\) unit ball of \(E\). It is a well-known fact, due to Pelczyński and Rolewicz, that the latter ``convex transitivity'' condition implies that \(E\) is a Hilbert space if \(\dim E<\infty\). See Proposition 9.6.1 and Theorem 9.7.1 in a book by \textit{S. Rolewicz} [Metric linear spaces, second ed. (1984; Zbl 0573.46001)]. The second part of the paper consists of a study of special subgroups of \(G\) in case that \(E=\{T:\mathbb{R}\to\mathbb{R}^ 2\mid\) \(T\) linear\(\}\) endowed with the operator norm.