The p-center problem with respect to a metric

$\rho $ (on the plane) consists in producing p points for a given set of n points to minimize the maximal distance (in the metric

$\rho )$ from the given points to their respective nearest produced points. The similar p-median problem is to minimize the sum of the considered distances. The main result of the paper states that both, p-center and p-median problem, are NP-hard for two metrics

$\rho ={\ell}^{2},{\ell}^{1}$. It is also proved that the p- center problem is NP-hard even if approximating it within 15 % for

$\rho ={\ell}^{2}$ and correspondingly within 50 % for

$\rho ={\ell}^{1}$.