An inequality on paths in a grid (Q1200004)

An inequality on paths in a grid is given in the paper. Let \((c_{i,j})\) be a positive \(n\times m\) log supermodular matrix, i.e., \(c_{i,j}\cdot c_{i+1,j+1}\geq c_{i+1,j}\cdot c_{i,j+1}\). Then there exists a path from \(c_{1,1}\) to \(c_{n,m}\) whose average dominates that of all the entries of the matrix \((c_{i,j})\).