generalized Ky Fan maximum principle in weakly compact JB$^*$-triples. More concretely, $\ell_p$-norms ($p\geq 1$) of the first $n$ singular values of an element are geometrically characterized in Theorem \[t Ky Fan maximum principle\].

The problem of the continuity of the spectral resolutions is treated in Section \[sec: perturbation of eigenvectors\]. If we consider the singular values of an element arranged in decreasing order but not counting multiplicity, for every singular value we get a finite-rank tripotent (the spectral resolution associated to a point). By Weyl inequality, given a (conveniently small) perturbation of the element, for each of this finite-rank tripotents we can associate another tripotent of the same rank being an spectral resolution of the perturbation. In Theorem \[t continuity of spectral resolutions\] it is shown the continuity of the associated spectral resolutions. This result provides a generalization of the already mentioned results of Davis and Becerra and Fernández-Polo to the setting of weakly compact JB$^*$-triples.

In the last section of this manuscript we will show that every weakly compact JB$^*$-triple satisfy the property $\hbox{\rm{(co)}}$ (see definition \[d (co)\]), that is, roughly speaking, perturbations of a convex combination of elements in its closed unit ball coincide with a convex combination 