The paper is devoted to the analysis of optimal simplicial meshes which minimize the gradient error of the piecewise linear interpolation over all conformal simplicial meshes with a fixed number of cells N T . We present theoretical results on asymptotic dependencies of L p -norms of the gradient error on N T for spaces of arbitrary dimension d. Our analysis is based on a geometric representation of the gradient error of linear interpolation on a simplex and a relaxed saturation assumption. We derive a metric field M p such that a M p -quasi-uniform mesh is quasi-optimal, for arbitrary d and p∊]0, +∞]. Quasi-optimal meshes provide the same asymptotics of the L p -norm of the gradient error as the optimal meshes.