  We study two subclasses of Banach limits: the one consisting of Banach limits
which are invariant with respect of the Ces\`aro operator and another one
consists of Banach limits which are invariant with respect to all dilations. We
prove that the first is a proper subset of the second. We also show that these
classes are at the maximal distance from the set ${\rm ext} \mathfrak B$ of all
extreme points of the set of all Banach limits.
