On \(\kappa\)-monotone couples of finite dimensional spaces (Q2780463)

Let \(w=\{w_i\}^n_{i=1}\) be such that \(w_1\geq w_2\geq \cdots\geq 0\). The Lorentz norm on \(\mathbb{R}^n\) is defined by \(\|x\|= \sum^n_{i=1} x^*_iw_i\), where \(\{x^*_i\}^n_{i=1}\) is the non-increasing rearrangement of \(\{|x_i|\}^n_{i=1}\). The Lorentz space \(\ell_n(w)\) is \(\mathbb{R}^n\) endowed with this norm. As usual \(\ell^n_\infty\) denotes \(\mathbb{R}^n\) with the supremum norm. The author obtains the formula for the \({\mathcal K}\)-functional of the couple \((\ell_n(w), \ell^n_\infty)\), namely: NEWLINE\[NEWLINE{\mathcal K}(t,x, \ell_n(w), \ell^n_\infty) =\begin{cases} tx^*_1, \quad & 0<t\leq \delta_1,\\ \Delta_i+ (t-\delta_i) x^*_{i+1},\quad & \Delta_i \leq t\leq \delta_{i+1},\\ \Delta_n,\quad & t\geq t_0,\end{cases}NEWLINE\]NEWLINE where \(\delta_i= \sum^i_{j=1}w_j\) and \(\Delta_i= \sum^i_{j=1} x^*_jw_j\). Using this formula, the author proves that if \((\ell_n(w), \ell^n_\infty)\) is an exact \({\mathcal K}\)-monotone couple, then \(w_n>0\).