We show that if m is a centered log-concave probability measure on Rn then, [(c1)/(vn)] = |Y2(m)|1/n = c2[(v{logn})/(vn)], where Y2(m) is the y2-body of m, and c1, c2 > 0 are absolute constants. It follows that m has "almost subgaussian" directions: there exists q ? Sn-1 such that m({ x ? Rn : |<x, q>| = c t E |<·, q>| } ) = e- [( t2)/(log(t+1))] for all 1 = t = v{nlogn}, where c > 0 is an absolute constant.