A dual capacitary Brunn-Minkowski inequality is established for the (n − 1)capacity of radial sums of star bodies in R. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in R, 1 ≤ p < n, proved by Borell, Colesanti, and Salani. When n ≥ 3, the dual capacitary BrunnMinkowski inequality follows from an inequality of...

1.1 Brunn-Minkowski inequality 1.1 Theorem. (Brunn-Minkowski, ’88) If A and B are non-empty compact sets then for all λ ∈ [0, 1] we have vol ((1− λ)A+ λB) ≥ (1− λ)(volA) + λ(volB). (B-M) Note that if either A = ∅ orB = ∅, this inequality does not hold since (1−λ)A+λB = ∅. We can use the homogenity of volume to rewrite Brunn-Minkowski inequality in the form vol (A+B) ≥ (volA) + (volB). (1.1) We ...

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a Hölder inequality, Minkowski convolution convolution-Hölder type inequality stability theorem to case the setting subspace Our unify refine existing literature.

we derive whole series of new integral inequalities of the hardy-type, with non-conjugate exponents. first, we prove and discuss two equivalent general inequa-li-ties of such type, as well as their corresponding reverse inequalities. general results are then applied to special hardy-type kernel and power weights. also, some estimates of weight functions and constant factors are obtained. ...

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then solve some related isoperimetric type problems convex hypersurfaces, which lead to new Alexandrov–Fenchel inequalities. particular, $n = 2$ obtain a Minkowski-type inequality and 3$ an optimal Willmore-type inequality. To prove these estimates, employ specifica...

– We present a one-dimensional version of the functional form of the geometric Brunn-Minkowski inequality in free (noncommutative) probability theory. The proof relies on matrix approximation as used recently by P. Biane and F. Hiai, D. Petz and Y. Ueda to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered ...