We present an argument which leads from the Brunn-Minkowski inequality to a Poincaré type inequality on the boundary of a convex body K of class C + in R . We prove that for every ψ ∈ C(∂K)

A description of continuous rigid motion compatible Minkowski valuations is established. As an application we present a Brunn–Minkowski type inequality for intrinsic volumes of these valuations.

The Brunn-Minkowski type and the cyclic inequalities for i- th general L p -mixed width-integral of convex bodies are established. Further, two differences obtained.

As Schneider [50] observes, the classical Brunn-Minkowski theory had its origin at the turn of the 19th into the 20th century, when Minkowski joined a method of combining convex bodies (which became known as Minkowski addition) with that of ordinary volume. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is that of projection body (precise definitions to fol...

In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.

In this study, the Brunn-Minkowski inequality for boxes is studied and a sharper version of derived by performing results based on abstract convexity.

For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equiv...

We prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex.

Week 1 (9/7/2010) . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Results and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Existence of Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . ...

We prove a functional version of the Brunn-Minkowski inequality for restricted sums obtained by Szarek and Voicu-lescu. We only consider Lebesgue-measurable subsets of R n , and for A ⊂ R n , we denote its volume by |A|. If A, B ⊂ R n , their Minkowski sum is defined by A + B = {x + y, (x, y) ∈ A × B}. The classical Brunn-Minkowski inequality provides a lower bound for its volume. In their stud...