A definition of oblique projections onto closed convex sets that
use seminorms induced by diagonal matrices which may have zeros
on the diagonal is introduced. Existence and uniqueness of such projections are secured via directional affinity of the sets with respect to the diagonal
matrices involved. A block-iterative algorithmic scheme for solving the convex feasibility problem, employing seminorm-induced oblique projections, is constructed and its
convergence for the consistent case is established. The fully simultaneous algorithm converges also in the inconsistent case to the minimum of a certain proximity function.