Geodesic
Approximation of Continuous Set-Valued Maps by Constant Set-Valued Maps with Image Balls
Matematičeskie zametki, Tome 82 (2007) no. 4, pp. 525-529.

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It is shown that the problem of the best uniform approximation in the Hausdorff metric of a continuous set-valued map with finite-dimensional compact convex images by constant set-valued maps whose images are balls in some norm can be reduced to a visual geometric problem. The latter consists in constructing a spherical layer of minimal thickness which contains the complement of a compact convex set to a larger compact convex set.
Keywords: set-valued map, approximation of set-valued maps, Hausdorff metric, subdifferential, compact convex set. 
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     author = {S. I. Dudov and A. B. Konoplev},
     title = {Approximation of {Continuous} {Set-Valued} {Maps} by {Constant} {Set-Valued} {Maps} with {Image} {Balls}},
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S. I. Dudov; A. B. Konoplev. Approximation of Continuous Set-Valued Maps by Constant Set-Valued Maps with Image Balls. Matematičeskie zametki, Tome 82 (2007) no. 4, pp. 525-529. http://geodesic.mathdoc.fr/item/MZM_2007_82_4_a6/

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