The concept of duality for measure projections of convex bodies

We show that an involution T on some class of functions on Rn, which reverses order (meaning that if f⩽g then Tf⩾Tg) has, often, a very specific form, actually essentially unique. It is done in this paper for the class of s-concave functions, for which this unique formula is derived. These functions are, for integer s, exactly marginals of convex bodies of dimension n+s. This understanding is also extended and discussed for other classes of functions, and represents from our point of view the abstract description of the concept of duality.