Geodesic
A Generalization of the Set Averaging Theorem
Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 410-416

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We consider the possibility of generalizing the averaging theorem from the case of sets from $n$-dimensional Euclidean space to the case of sets from Banach spaces. The result is a cornerstone for constructing the theory of the Riemann integral for non-convex-valued multivalued mappings and for proving the convexity of this multivalued integral. We obtain a generalization of the averaging theorem to the case of sets from uniformly smooth Banach spaces as well as some corollaries.
Keywords: set averaging theorem, $n$-dimensional Euclidean space, Banach space, Riemann integral, non-convex-valued multivalued mapping, convex compact set, Hausdorff metric. 
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     author = {G. Ivanov and E. S. Polovinkin},
     title = {A {Generalization} of the {Set} {Averaging} {Theorem}},
     journal = {Matemati\v{c}eskie zametki},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a8/}
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G. Ivanov; E. S. Polovinkin. A Generalization of the Set Averaging Theorem. Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 410-416. http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a8/