CHARACTERIZATION OF CONTINUOUS ADDITIVE SET-VALUED MAPS "MODULO K" ON FINITE DIMENSIONAL LINEAR SPACES

Let Y be a real vector metric space and K subset of Y be a closed convex cone such that K boolean AND (-K) K ) = {0}. We prove that a convex compact-valued map F : R -> 2(Y) Y \ {& empty;} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B subset of Y such that 0 is an element of A - B subset of K and F is equal "modulo K " to the continuous set-valued map tA, t >= 0 , G ( t ) = tB, t < 0 . Next, we use this result to characterize convex compact-valued maps F : R N -> 2Y Y \ {& empty;}.