Least Deviation Decomposition with Respect to a Pair of Convex Sets

Let $K_1$ and $K_2$ be two nonempty closed convex sets in some normed space $(H,\|\cdot\|)$. This paper is concerned with the question of finding a "good" decomposition, with respect to $K_1$ and $K_2$, of a given element of the Minkowski sum $K_1+K_2$. We introduce and discuss the concept of least deviation decomposition. This concept is an extension of the Moreau orthogonal decomposition with respect to a pair of mutually polar cones. Techniques of convex analysis are applied to obtain some sensitivity and duality results related to the decomposition problem.