Two characterizations of smooth norms
Let \(\gamma\) be a norm with unit ball \(B = \{x\in\mathbb R^n: \gamma(x)\leqslant 1\}\), \(n\geqslant 2.\) The norm \(\gamma\) and its unit ball \(B\) are said to be smooth if for each \(t\in\mathbb R^n\) with \(\gamma(t) = 1\) there exists exactly one supporting hyperplane of \(B\) at \(t.\) The authors give the characterizations of smooth norms. The norm \(\gamma\) is smooth iff for all weighted point sets \(P\) not contained in a hyperplane each median hyperplane (with respect to \(\gamma\) and \(P\)) is the affine hull of \(n\) points from \(P,\) i.e., each median hyperplane satisfies the incidence criterion. The norm \(\gamma\) is smooth iff for all weighted point sets \(P\) not contained in a hyperplane each center hyperplane (with respect to \(\gamma\) and \(P\)) is at maximum distance to \(n + 1\) affinely independent points from \(P,\) i.e., every center hyperplane is blocked by \(P.\)
scientific article; zbMATH DE number 1347334