Аннотация:Let X be a real linear space and B is a bounded convex subset of X. Consider the distance function ρ B (x,y)=ρ(0,y-x), where ρ(0,x)=ρ(x) is the Minkowski functional relative to B. A subset M⊂X is called: a Chebyshev set relative to ρ B if P M x={y∈M|ρ B (x,y)=ρ B (x,M)} is a single-point set for every x∈X; M is called a sun relative to ρ B if for each point x∉M there is a point y∈P M x such that y∈P M z for every z in the ray initiating at y and containing x. For a two-dimensional linear space X there are found necessary and sufficient conditions for a bounded convex set B⊂X and for a compact subset K of ℝ 2 so that K can be embedded into X as a Chebyshev set or as a sun relative to ρ B . The result answers partially a question posed by S. V. Konyagin.