For a graph \(G=(V,E)\) a function \(g:V\to [0,1]\) is a fractional dominating function if the sum of the values of \(g\) over the closed neighborhood of every vertex is at least \(1\). The authors study the minimum \(\ell^p\) norm for \(1\leq p\leq \infty\) of the vectors \((g(u))_{u\in V}\) over the set of fractional dominating functions of a graph. From standard convexity arguments they derive the uniqueness of this minimum for \(1<p<\infty\), its continuity and monotonicity as a function of \(p\) and the convergence of the optimal fractional dominating functions for \(p\to 1\) and \(p\to\infty\).