A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre-Fenchel conjugates for set-valued functions is introduced and a Moreau-Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.