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
Received March 17, 2025
Revised April 9, 2025
Accepted April 14, 2025
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2025-1549).
Vladimir Nikolaevich Ushakov, Corresponding Member of the Russian Academy of Sciences, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: ushak@imm.uran.ru
Aleksandr Anatol’evich Ershov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108, Russia, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: ale10919@yandex.ru
Oleg Aleksandrovich Kuvshinov, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: okuvshinov@inbox.ru
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Cite this article as: V.N. Ushakov, A.A. Ershov, O.A. Kuvshinov. On the area of the $\varepsilon$-layer of a weakly convex figure. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 280–293.
