  We introduce and study the class of extremally rich JB$^*$-triples. We
establish new results to determine the distance from an element $a$ in an
extremally rich JB$^*$-triple $E$ to the set $\partial_{e} (E_1)$ of all
extreme points of the closed unit ball of $E$. More concretely, we prove that
$$\hbox{dist} (a,\partial_e (E_1)) =\max \{ 1, \|a\|-1\},$$ for every $a\in E$
which is not Brown-Pedersen quasi-invertible. As a consequence, we determine
the form of the $\lambda$-function of Aron and Lohman on the open unit ball of
an extremally rich JB$^*$-triple $E$, by showing that $\lambda (a)= \frac12$
for every non-BP quasi-invertible element $a$ in the open unit ball of $E$. We
also prove that for an extremally rich JB$^*$-triple $E$, the quadratic connorm
$\gamma^{q}(.)$ is continuous at a point $a\in E$ if, and only if, either $a$
is not von Neumann regular {\rm(}i.e. $\gamma^{q}(a)=0${\rm)} or $a$ is
Brown-Pedersen quasi-invertible.
