Geometric relation between centrality and the impact parameter in relativistic heavy-ion collisions

We show, under general assumptions which are well satisfied in relativistic heavy-ion collisions, that the geometric relation of centrality c to the impact parameter b, namely c ~ pi b^2/sigma_inel, holds to a very high accuracy for all but most peripheral collisions. More precisely, if c(N) is the centrality of events with the multiplicity higer than N, then b is the value of the impact parameter for which the average multiplicity <n(b)> is equal to N. The corrections to this geometric formula are of the order (Delta n(b)/<n(b)>)^2, where Delta n(b) is the width of the multiplicity distribution at a given value of b, hence are very small. In other words, the centrality effectively measures the impact parameter.