A combinatorial characterisation of amenable locally compact groups

We introduce a new combinatorial condition that characterises the amenability for locally compact groups. Our condition is weaker than the well-known F{\o}lner's conditions, and so is potentially useful as a criteria to show the amenability of specific locally compact groups. Our proof requires us to give a quantitative characterisation of (relatively) weakly compact subsets of ${\mathfrak L}^1$-spaces, and we do this through the introduction of a new notion of almost $(p,q)$-multi-boundedness for a subset of a Banach space that is intimately related to the well-known notion of the $(q,p)$-summing constants of an operator. As a side product, we also obtain a characterisation of weakly compact operators from ${\mathfrak L}^\infty$-spaces in terms of their sequences of $(q,p)$-summing constants.