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Extreme symmetric norms on \(R^ 2\). (English) Zbl 0671.46006

Summary: By a norm on \({\mathbb{R}}^ 2\) we mean a functional N: \({\mathbb{R}}^ 2\to {\mathbb{R}}_+\) such that
1) \(N(u)>0\) for \(u\neq 0,\)
2) \(N(u+v)\leq N(u)+N(v),\)
3) \(N(\lambda u)=| \lambda | N(u)\), \(\lambda\in {\mathbb{R}}.\)
A norm N is symmetric if \(N((x,y))=N((| x|,| y|))\). We denote by \({\mathcal S}\) the set of all symmetric norms on \({\mathbb{R}}^ 2\) satisfyng the condition \(N((1,0))=N(0,1))=1\). The set \({\mathcal S}\) is convex. The purpose of this note is to give a characterization of the set of extreme points of \({\mathcal S}\) (denoted by ex \({\mathcal S})\). This solves the problem (P 1223) posed by Professor A. Pietsch at the Winter School on Functional Analysis in January 1978 [see N. Tomczak-Jaegermann, Colloq. Math. 45, 45-47 (1981; Zbl 0495.46016)].

MSC:

46B20 Geometry and structure of normed linear spaces
46B03 Isomorphic theory (including renorming) of Banach spaces

Citations:

Zbl 0495.46016
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