  We prove a new Bernstein type inequality in $L^p$ spaces associated with the
tangential derivatives on the boundary of a general compact $C^2$-domain. We
give two applications: Marcinkiewicz type inequality for discretization of
$L^p$ norm and positive cubature formula. Both results are optimal in the sense
that the number of function samples used has the order of the dimension of the
corresponding space of algebraic polynomials.
