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Journal of Convex Analysis 20 (2013), No. 1, 025--042
Copyright Heldermann Verlag 2013

Smooth Selections of Convex-Valued Multifunctions

Jacek Sadowski
Faculty of Mathematics, Politechnika Warszawska, ul. Koszykowa 75, 00-662 Warszawa, Poland


We establish a class of multifunctions having smooth ($C^\infty$) selections and formulate assumptions on a multifunction $F$ under which for any continuous selection $f$ of $F$ there is a~sequence of smooth selections of $F$ converging uniformly to $f$. Moreover, we obtain a Castaing type representation of multifunctions by a sequence of smooth selections, i.e. we construct a sequence $\{f_k\}$ of smooth selections of $F$ satisfying the condition $F(x)=\overline{\cup_{k\geq 1} \ f_k(x)}$ for all $x\in X$.

Keywords: Lower semicontinuous multifunction, smooth selection, uniform convergence, approximation, convolution, Castaing representation.

MSC: 26E25, 54C60, 54C65

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