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.

The Schauder conjecture that every compact convex subset of a metric linear space has the fixed-point property was recently established by Cauty (2001). This paper elaborates on Cauty's proof in order to make it more detailed, and therefore more accessible. Such a detailed analysis allows us to show that the convex compacta in metric linear spaces possess the simplicial approximation property introduced by Kalton, Peck, and Roberts. The latter demonstrates that the original Schauder approach to solve the conjecture is in some sense “correctable.”

We establish Birkhoff-Kellogg type theorems on invariant directions for a general class of maps. Our results, in particular, apply to Kakutani, acyclic, O'Neill, approximable, admissible, and ${𝒰}_c^{\kappa }$ maps.