$\mathcal{U}$, a tripotent $e$ in $\mathcal{U}$ and a minimal tripotent $v$ in $ \mathcal{U}_2(e) \cup \mathcal{U}_0(e)$ it should be clear that $\|P_2(e)x\|_v, \|P_0(e)x\|_v \leq \|x\|_v$. Therefore a direct application of the minimax theorem allows us to state the following Interlacing theorem or Cauchy-Poincaré theorem for weakly compact JB$^*$-triples (see for example [@Bha Theorem 2.1] for the matricial case).

\[t Interlacing theorem of Cauchy-Poincare\] Let $\mathcal{U}$ be a weakly compact JB$^*$-triple. Let $x$ be an element in $\mathcal{U}$ with singular values $(\lambda_n(x))_{n\in \mathbb{N}}$. Then for every tripotent $e$ in $\mathcal{U}$ each singular value of $P_2(e)x$ and $P_0(e)x$ is bounded by the corresponding singular value of $x$, i.e. $$\lambda_n(P_2(e)x), \lambda_n(P_0(e)x) \leq \lambda_n(x).$$ $\hfill\Box$

We finish this section by presenting another characterization of the $n$-th singular value of an element in terms of the distance to the set of elements of rank $n$ (see [@GohKre Chapter II. Theorem 2.1] for compact operators on a Hilbert space). When particularized to the space of compact operators on a Hilbert space, which is a weakly compact JB$^*$-triple (every compact C$^*$-algebra is), we get exactly the just quoted result.

\[t other characterization of singular values\] Let $\mathcal{U}$ be a weakly compact JB$^*$-triple, $x\in  \mathcal{U}$. Then $$\lambda_{n}(x)=\min \{\|x-a\|: 