On maximal relative projection constants

This article focuses on the maximum of relative projection constants over all m-dimensional subspaces of the N-dimensional coordinate space equipped with the max-norm. This quantity, called maximal relative projection constant, is studied in parallel with a lower bound, dubbed quasimaximal relative projection constant. Exploiting alternative expressions for these quantities, we show how they can be computed when N is small and how to reverse the Kadec–Snobar inequality when N   does not tend to infinity. Precisely, we first prove that the (quasi)maximal relative projection constant can be lower-bounded by cm, with c arbitrarily close to one, when N is superlinear in m  . The main ingredient is a connection with equiangular tight frames. By using the semicircle law, we then prove that the lower bound cm holds with c<1c<1 when N is linear in m.