An optimal lower bound for the Frobenius problem

Given N⩾2 positive integers a1,a2,…,aN with GCD(a1,…,aN)=1, let fN denote the largest natural number which is not a positive integer combination of a1,…,aN. This paper gives an optimal lower bound for fN in terms of the absolute inhomogeneous minimum of the standard (N−1)-simplex.