Geodesic
Lipschitz continuous parametrizations of set-valued maps with
Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1123-1143

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We continue the investigations started in [1]–[4], where weakly convex sets and set-valued maps with weakly convex images were studied. Sufficient conditions are found for the existence of a Lipschitz parametrization for a set-valued map with solidly smooth (generally, non-convex) images. It is also shown that the set-valued $\varepsilon$-projection on a weakly convex set and the unit outer normal vector to a solidly smooth set satisfy, as set functions, the Lipschitz condition and the Hölder condition with exponent $1/2$, respectively, relative to the Hausdorff metric. 
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     title = {Lipschitz continuous parametrizations of set-valued maps with},
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G. E. Ivanov; M. V. Balashov. Lipschitz continuous parametrizations of set-valued maps with. Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1123-1143. http://geodesic.mathdoc.fr/item/IM2_2007_71_6_a2/