Extreme lower previsions

Coherent lower previsions constitute a convex set that is closed and compact under the topology of point-wise convergence, and Maaß [13] has shown that any coherent lower prevision can be written as a 'countably additive convex combination' of the extreme points of this set. We show that when the possibility space has a finite number n of elements, these extreme points are either degenerate precise probabilities, or fully imprecise and in a one-to-one correspondence with Minkowski indecomposable non-empty convex compact subsets of Rn−1. By exploiting this connection, we are able to prove that for n=3, fully imprecise extreme lower previsions are lower envelopes of at most three linear previsions. For n≥4, 'most' fully imprecise lower previsions are extreme. Finally, we show that in our context, Maaß's result can be strengthened as follows: any coherent lower prevision can be written as, or approximated arbitrarily closely by, a finite convex combination of finitely generated extreme lower previsions.