On Linear Selections of Convex Set-Valued Maps

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.