The concept of duality for measure projections of convex bodies

We show that an involution T on some class of functions on R-n, which reverses order (meaning that if f <= g then T f >= T g) 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. (c) 2007 Elsevier Inc. All rights reserved.