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Journal of Convex Analysis 27 (2020), No. 4, 1123--1136
Copyright Heldermann Verlag 2020

Nonlinear Images of Sets. II: Applications to Differential Games

Grigorii E. Ivanov
Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Moscow Region, Russia

This paper is a continuation of our previous study on weak convexity of nonlinear images of sets [see G. E. Ivanov, Nonlinear images of sets. I: Strong and weak convexity, J. Convex Analysis 27(1) (2020) 363--382]. We consider smoothness of a set in a special sense called "bodily smoothness" and split this smoothness property of a set S into two properties: weak convexity of S and weak convexity of the closure of the complementary to S. We obtain sufficient conditions for bodily smoothness of the image of the Cartesian product of two sets S × U, wherein S is bodily smooth and U is strongly convex. We apply this result for nonlinear differential games and obtain sufficient conditions for the game reachable sets to be smooth.

Keywords: C1,1 mapping, covering mapping, bi-Lipschitz mapping, strong convexity, weak convexity, Minkowski sum, Minkowski operator, differential game.

MSC: 52A05, 46T20, 49N70.

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