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2014•Bulletin of the Australian Mathematical SocietyOpen access

A FURTHER PROPERTY OF SPHERICAL ISOMETRIES

Ryotaro Tanaka

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Abstract

Abstract In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and $Y$ are $n$ -dimensional Banach spaces and $T_0$ is an isometry from the unit sphere of $X$ onto that of $Y$ then it maps the set of all $(n-1)$ -extreme points of the unit ball of $X$ onto that of $Y$ .

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Abstract In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and $Y$ are $n$ -dimensional Banach spaces and $T_0$ is an isometry from the unit sphere of $X$ onto that of $Y$ then it maps the set of all $(n-1)$ -extreme points of the unit ball of $X$ onto that of $Y$ .

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Available abstract

Abstract In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and $Y$ are $n$ -dimensional Banach spaces and $T_0$ is an isometry from the unit sphere of $X$ onto that of $Y$ then it maps the set of all $(n-1)$ -extreme points of the unit ball of $X$ onto that of $Y$ .

Key concepts: Unit sphere, Mathematics, Isometry (Riemannian geometry), Banach space, Unit (ring theory), Combinatorics, SPHERES, Ball (mathematics)

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