In their main result, the authors prove that every time we find an \(n\)-dimensional subspace \(X_n\) of \(l_\infty^N\) that has a maximal projection constant (usually denoted by \(\lambda(X_n)\)) among all \(n\)-dimensional spaces, then there exists an \(n\)-dimensional subspace \(Y_n\) of \(l_1^N\) also having maximal projection constant (that is, with \(\lambda(Y_n)=\lambda(X_n)\)). Correspondingly, a similar result holds for the so-called relative projection constants: if \(X_n\subseteq l_\infty^N\) is such that \(\lambda(X_n,l_\infty^N)\) is maximal among all \(n\)-dimensional subspaces of any \(N\)-dimensional space, then there exists a \(Y_n\subseteq l_1^N\) such that \(\lambda(Y_n,l_1^N)\) is also equal to this maximal value. To complement this result, using probabilistic methods the authors also show that there is an infinite amount of mutually non-isometric subspaces \(Y_n\) of \(l_1^N\) that would make the first part of the above theorem true. The authors also discuss several concrete examples in detail.