Abstract <p> For a class of closed nonconvex sets in the two-dimensional Euclidean space, we propose an approach to evaluating the Chebyshev layer based on two well-known concepts that generalize the definition of a convex set. We consider a family of planar sets with finitely many pseudovertices and select three sets of pseudovertices for analysis which differ in the order of smoothness of their pseudovertices. Within each of these three cases (the case of a piecewise smooth boundary of the set, the case of a discontinuity in the curvature of the boundary of the set, and the classical case, where the curvature of the boundary is continuous), we find a formula for the limit value of the radii of Efimov–Stechkin support balls. Specifically, we consider balls with centers on a branch of the bisector (a one-dimensional manifold of the nonuniqueness set) corresponding to the chosen pseudovertex. Using these formulas, we evaluate analytically the Chebyshev layers for nonconvex sets, including those with boundaries of variable smoothness. We also give an illustrative example and its interpretation from the viewpoint of optimal control theory. </p>