Thenumber of points where a linear mapping from l<sup>n</sup><sub>2</sub> into l<sup>n</sup><sub>p</sub> attains its norm


Let S be a regular n  n -matrix mapping l<sub>2</sub><sup>n</sup> onto l<sub>p</sub><sup>n</sup>, 1≤ p < ∈fty, with norm(Error rendering LaTeX formula). Then we are interested in the set (Error rendering LaTeX formula) i.e. the set of points on the unit sphere where S attains its norm. We prove card(C) <∈fty for 1≤ p < 2 .This follows from properties of the Taylor expansion of x → \|Sx\|<sub>p</sub> near points in C. The case 2 < p < ∈fty remains open. But we show by an example that for p> 2 the behaviour of x → \|Sx\|<sub>p</sub> may be completely different as for p < 2 .

DOI Code: 10.1285/i15900932v12p145

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