Let \({\mathbb{N}}\) be the set of all natural numbers and \({\ell_\infty=\ell_\infty (\mathbb{N})}\) be the Banach space of all bounded sequences x = (x1, x2 . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let \({\ell_\infty^*}\) be its Banach dual. Let \({\mathfrak{B} \subset \ell_\infty^*}\) be the set of all normalised positive translation invariant functionals (Banach limits) on l∞ and let \({ext(\mathfrak{B})}\) be the set of all extreme points of \({\mathfrak{B}}\) . We prove that an arbitrary sequence (Bj)j ≥ 1, of distinct points from the set \({ext(\mathfrak{B})}\) is 1-equivalent to the unit vector basis of the space l1 of all summable sequences. We also study Cesaro-invariant Banach limits. In particular, we prove that the norm closed convex hull of \({ext(\mathfrak{B})}\) does not contain a Cesaro-invariant Banach limit.