Geometric properties of the set of Banach limits

We study the geometry and extreme points of the set $\mathfrak B\subset l_\infty^*$ of all positive normalized shift-invariant functionals on the space $l_\infty$ of all bounded sequences with the uniform norm. In particular, we calculate the radius of $\mathfrak B$ and, for a large class of sequences $x$, describe the orbit of $x$ under the extreme points of $\mathfrak B$.