Thenumber of points where a linear mapping from $lⁿ₂$ into $lⁿ_p$ attains its norm

Abstract

Let S be a regular $n ? n$ -matrix mapping $l₂ⁿ$ onto $l_pⁿ$, $1≤ p < ∈fty$, with norm $\|S\|=\| S: l₂^n→ l_pⁿ\|$. Then we are interested in the set $$C := {z ∈ Rⁿ; \|z\|₂=1\;\textrm{and}\; \|Sz\|_p=\|S\|},$$ 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 $x → \|Sx\|_p$ near points in C. The case $2 < p < ∈fty$ remains open. But we show by an example that for $p> 2$ the behaviour of $x → \|Sx\|_p$ may be completely different as for $p < 2$ .