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.

In this paper we prove a series of Rogers-Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities, or quasi-concave densities attaining their maximum at the origin. Functional versions of classical Rogers-Shephard inequalities are also derived as consequences of our approach.

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.

Given one metric measure space X satisfying a linear Brunn-Minkowski inequality, and a second one Y satisfying a Brunn-Minkowski inequality with exponent p ≥ −1, we prove that the product X × Y with the standard product distance and measure satisfies a Brunn-Minkowski inequality of order 1/(1 + p −1 ) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn-Minkowski inequality is obtained in X × Y when Y satisfies a Prékopa-Leindler inequality.In particular, we show that the classical Brunn-Minkowski inequality holds for any pair of weakly unconditional sets in R n (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn-Minkowski inequality.Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn-Minkowski's inequalities are derived from our results.

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