Approximation and convex decomposition by extremals and the $\lambda$-function in JBW*-triples

We establish new estimates to compute the $\lambda$-function of Aron and Lohman on the unit ball of a JB$^*$-triple. It is established that for every Brown-Pedersen quasi-invertible element $a$ in a JB$^*$-triple $E$ we have $$\hbox{dist} (a, \mathfrak{E} (E_1)) = \max \left\{ 1- m_q (a) , \|a\|-1\right\},$$ where $\mathfrak{E} (E_1)$ denotes the set of extreme points of the closed unit ball $E_1$ of $E$. It is proved that $\lambda (a) = \frac{1+m_q (a)}{2},$ for every Brown-Pedersen quasi-invertible element $a$ in $E_1$, where $m_q (a)$ is the square root of the quadratic conorm of $a$. For an element $a$ in $E_1$ which is not Brown-Pedersen quasi-invertible we can only estimate that $\lambda (a)\leq \frac12 (1-\alpha_q (a)).$ A complete description of the $\lambda$-function on the closed unit ball of every JBW$^*$-triple is also provided, and as a consequence, we prove that every JBW$^*$-triple satisfies the uniform $\lambda$-property.