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 Folner'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 L1-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 L-infinity-spaces in terms of their sequences of (q,p)-summing constants.