Geodesic
On Linear Selections of Convex Set-Valued Maps
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 56-68

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We study continuous subadditive set-valued maps taking points of a linear space $X$ to convex compact subsets of a linear space $Y$. The subadditivity means that $\varphi(x_1+x_2)\subset \varphi(x_1) + \varphi(x_2)$. We characterize all pairs of locally convex spaces $(X, Y)$ for which any such map has a linear selection, i.e., there exists a linear operator $A\colon X \to Y$ such that $Ax \in \varphi (x)$, $x\in X$. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.
Keywords: set-valued map, linear selection, subadditivity, Lipschitz function, stability of linear operators. 
@article{FAA_2011_45_1_a4,
     author = {V. Yu. Protasov},
     title = {On {Linear} {Selections} of {Convex} {Set-Valued} {Maps}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {56--68},
     publisher = {mathdoc},
     volume = {45},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a4/}
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V. Yu. Protasov. On Linear Selections of Convex Set-Valued Maps. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 56-68. http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a4/