The best asymmetric approximation in spaces of continuous functions

We consider approximation by convex sets in the space of continuous maps from a compact topological space to a locally convex space with respect to certain asymmetric seminorms. We suggest new criteria for elements of least deviation, make a definition of strongly unique elements of least deviation and study the problems of characterization and existence of such elements. The most detailed study concerns the approximation with a sign-sensitive weight of real-valued continuous functions defined on a compact metric space or on a line segment by elements of the Chebyshev space.