  Given an integer mxn matrix A satisfying certain regularity assumptions, we
consider the set F(A) of all integer vectors b such that the associated
knapsack polytope P(A,b)={x: Ax=b, x>=0} contains an integer point. When m=1
the set F(A) is known to contain all consecutive integers greater than the
Frobenius number associated with A. In this paper we introduce the diagonal
Frobenius number g(A) which reflects in an analogous way feasibility properties
of the problem and the structure of F(A) in the general case. We give an
optimal upper bound for g(A) and also estimate the asymptotic growth of the
diagonal Frobenius number on average.
