ABSTRACT. We consider a functional F on the space of convex bodies in R n of the form) is a given continuous function on the unit sphere of R n , K is a convex body in R n , n ≥ 3, and Sn−1(K, ⋅) is the area measure of K. We prove that F satisfies an inequality of Brunn-Minkowski type if and only if f is the support function of a convex body, i.e., F is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n − 1 and satisfy a Brunn-Minkowski type inequality.

As a substraction counterpart of the well-known p-sum of convex bodies, we introduce the notion of p-difference. We prove several properties of the p-difference, introducing also the notion of p-(inner) parallel bodies. We prove an analog of the concavity of the family of classical parallel bodies for the p-parallel ones, as well as the continuity of this new family, in its definition parameter. Further results on inner parallel bodies are extended to p-inner ones; for instance, we show that tangential bodies are characterized as the only convex bodies such that their p-inner parallel bodies are homothetic copies of them.

The Rogers–Shephard and Brunn–Minkowski inequalities provide upper and lower bounds for the volume of the difference body in terms of the volume of the body itself. In this work it is shown that the difference body operator is the only continuous and

Abstract. If f , g : R n −→ R ≥0 are non-negative measurable functions, then the Prékopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the 0-mean of the integrals of f and g. In this paper we prove that under the sole assumption that f and g have a common projection onto a hyperplane, the Prékopa-Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.

For a broad class of integral functionals defined on the space of n-dimensional convex bodies, we establish necessary and sufficient conditions for monotonicity, and necessary conditions for the validity of a Brunn-Minkowski type inequality. In particular, we prove that a Brunn-Minkowski type inequality implies monotonicity, and that a general Brunn-Minkowski type inequality is equivalent to the functional being a mixed volume.