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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   \[{\rm dist} (a, \mathfrak {E} (E_1)) = \max \{ 1- m_q (a), \|a\|-1\} ,\]   where   $\mathfrak {E} (E_1)$   denotes the set of extreme points of the closed unit ball   $E_1$   of   $E$  . It is proved that   $\lambda (a) = ({1+m_q (a)})/{2},$   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   $\lambda (a)\leq \frac 12 (1-\alpha _q (a))$  . 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.

APPROXIMATION AND CONVEX DECOMPOSITION BY EXTREMALS AND THE λ-FUNCTION IN JBW*-TRIPLES