On the area of the $\varepsilon$-layer of a weakly convex figure

V. N. Ushakov, A. A. Ershov, O. A. Kuvshinov

N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: The article is devoted to the generalization of the well-known Steiner formula for the volume of the $\varepsilon$-neighborhood of a convex body in $n$-dimensional Euclidean space to some classes of nonconvex bodies. This study is limited to the case of two-dimensional Euclidean space, with flat figures located in it and their neighborhoods. Examples of various nonconvex figures on the plane are considered, for the neighborhood of which the Steiner formula is either satisfied or not satisfied. The Steiner formula for calculating the area of the $\varepsilon$-layer of weakly convex Efimov–Stechkin plane figures with a smooth boundary is substantiated. The proof is based on methods of differential geometry and properties of weakly convex sets.

Keywords: Steiner formula, nonconvex figure, area of neighborhood, parallel body, weakly convex set.

UDC: 514.752.3

MSC: 51N20

Received: 17.03.2025
Revised: 09.04.2025
Accepted: 14.04.2025

DOI: 10.21538/0134-4889-2025-31-2-280-293

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