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 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. © 2007 Elsevier Inc. All rights reserved.