Abstract
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 \(\varepsilon\) -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 \(\varepsilon\) -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.
