with equality if and only if f and g are proportional. For p <0, we assume that f(x), g (x) >0. An (almost) improvement of Minkowski’s inequality, for p Î R\{0}, is obtained in the following Theorem: Theorem 1.2 Let f(x), g(x) ≥ 0 and p >0, or f(x), g(x) >0 and p <0. Let s, t Î R\{0}, and s ≠ t. Then (i) Let p, s, t Î R be different, such that s, t >1 and (s t)/(p t) >1. Then ∫ (f (x)+g(x))pdx ...

The above norm induces a metric d where d(f, g) = ‖f − g‖p. Note that d(f, g) = 0 if and only if f = g a.e. μ, in which case we identify f with g. The Lp norm, like all worthy norms, satisfies the triangle inequality: ‖f + g‖p ≤ ‖f‖p + ‖g‖p ; this is precisely Minkowski’s inequality. For random variables X, Y defined on the same probability space and having finite p’th moments, Minkowski’s ineq...

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.

Moreover, we introduce a curvature-dimension condition CD(K, N) being more restrictive than the curvature bound Curv(M,d, m) ≥ K. For Riemannian manifolds, CD(K, N) is equivalent to RicM (ξ, ξ) ≥ K · |ξ|2 and dim(M) ≤ N . Condition CD(K,N) implies sharp version of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows ...

In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global existence theorems. We show that global solutions exist for a certain class of collision cross sections of the hard potential type in Minkowski space and in ...

Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in On the similarity of the entropy power inequality and the BrunnMinkowski inequality (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the 1 n -concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We inve...

We study a Riemannian manifold equipped with a density which satisfies the Bakry–Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, origin...

Abstract: An affine Moser-Trudinger inequality, which is stronger than the Euclidean MoserTrudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L energy of gradient. The geometric inequality at the core of the affine Moser-Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies...

In this paper we study the impact of Minkowski metric matrix on a projection in the Minkowski Space M along with their basic algebraic and geometric properties.The relation between the m-projections and the Minkowski inverse of a matrix A in the minkowski space M is derived. In the remaining portion commutativity of Minkowski inverse in Minkowski Space M is analyzed in terms of m-projections as...