The article is devoted to a generalization to some classes of non-convex bodies of the well-known Steiner formula for the volume of the -neighborhood of a convex body in the n -dimensional Euclidean space. This study is limited to the case of the two-dimensional Euclidean space, flat figures located in it and their neighborhoods. Examples of various non-convex figures in the plane are considered for the neighborhood of which the Steiner formula is both satisfied and not satisfied. The Steiner formula for computing the area of the -layer of plane Efimov–Stechkin weakly convex figures with a smooth boundary is justified. The proof is based on methods of differential geometry and properties of weakly convex sets.
Ushakov et al. (Fri,) studied this question.