$v\in \mathcal{U}$.

Assume now that $n\geq 2$. Take $e$ an arbitrary tripotent in $\mathcal{U}$ of rank $n$. We have that $\{v_i: i=1,\ldots,n-1\}$ is a family of mutually orthogonal minimal tripotents and Corollary \[c gram schmidt\] assures the existence of a minimal tripotent $v$ in $\mathcal{U}_2(e)$ such that $v\in \{ v_1, \ldots, v_{n-1}\}^{\perp}$. Since $v\perp v_i$ we have that $\|v_i\|_{v}=0$ ($i=1,\ldots,n-1$) and $$\|x\|_{v}\leq \sum_{i=1}^{n-1} \|\lambda_i(x) v_i\|_{v} + \|\sum_{i\geq n}\lambda_i(x) v_i\|_{v}\leq \|\sum_{i\geq n}\lambda_i(x) v_i \|=\lambda_n(x).$$

The following result is the announced generalization of the minimax principle to the setting of weakly compact JB$^*$-triples.

\[t minmax\] Let $x$ be an element in a weakly compact JB$^*$-triple $\mathcal{U}$. Then for every natural $n$ we have $$\begin{aligned}
\label{eq minmax principle}\lambda_n(x) &=\max_e \inf \{\|x\|_{v}: v\in \mathcal{U}_2(e) \hbox{ minimal tripotent} \} \\ &=\min_f \sup\{ \|x\|_{v}: v\in \mathcal{U}_0(f) \hbox{ minimal tripotent}\} \notag \end{aligned}$$ where $e$ (respectively, $f$) runs over the set of tripotents in $\mathcal{U}$ of rank $n$ (respectively, $n-1$) and $\lambda_n(x)$ is the $n$-th singular value of $x$.

We will prove first that $\lambda_n(x) =\displaystyle\max_e \inf \{\|x\|_{v}: v\in \mathcal{U}_2(e) \hbox{ minimal tripotent} \}$ where $e$ runs over the set of tripotents in $\mathcal{U}$ of rank $n$. To this end, fix an atomic decomposition of the element $x=\sum 