In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended capacities constructed authors based on a class Hamiltonian non-periodic boundary value problems recently. Then introduce billiards domains, and them prove some corresponding results those per...

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving ϕ $$ \phi -Laplacian operators in a ball. In particular, p Minkowski-curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to Harnack-type inequality terms seminorm. As consequence result, it is also derived several (even infinitely many)...

Digital subscriber lines (DSL) are fundamentally limited by crosstalk. The case where all crosstalk is from the same type of DSL has been studied over the years and accurate models have been standardized. However, crosstalk from multiple different types of DSLs is a relatively new area of study and models of summing mixed crosstalk have only recently been postulated. In this contribution, a new...

The Brunn-Miknowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This classical inequality in convex geometry was inspired by issues around the isoperimetric problem and was considered for a long time to belong to geometry, where its significance is widely recognized. However, it is by now clear that the Brunn-Miknowski ineq...

We initiate a systematic investigation into the nature of the function αK (L, ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK (L, ρ) that gives the (n − 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρL) of ρL. The focus is on the concavity properties of αK (L, ρ). ...

We prove a quantitative stability result for the Brunn-Minkowski inequality: if |A| = |B| = 1, t ∈ [τ, 1−τ ] with τ > 0, and |tA+(1−t)B| ≤ 1+δ for some small δ, then, up to a translation, both A and B are quantitatively close (in terms of δ) to a convex set K.

Example 1.2. a) Let X = R and take d(x, y) = |x − y|. This is the most basic and important example. b) More generally, let N ≥ 1, let X = R , and take d(x, y) = ||x − y|| = √∑N i=1(xi − yi). It is very well known but not very obvious that d satisfies the triangle inequality. This is a special case of Minkowski’s Inequality, which will be studied later. c) More generally let p ∈ [1,∞), let N ≥ 1...

Abstract The aim of this work is to obtain quantum estimates for q -Hardy type integral inequalities on calculus. For this, we establish new identities including derivatives and numbers. After that, prove a generalized -Minkowski inequality. Finally, with the help obtained equalities inequality, results want. outcomes presented in paper are -extensions -generalizations comparable literature ine...

The arithmetic-geometric mean inequality in short, AG inequality has been widely used in mathematics and in its applications. A large number of its equivalent forms have also been developed in several areas of mathematics. For probability and mathematical statistics, the equivalent forms of the AG inequality have not been linked together in a formal way. The purpose of this paper is to prove th...