Refinements of the Converse Hölder and Minkowski Inequalities

We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the converse Minkowski inequality for functionals and of its continuous form and give refinements of it. Application on integral mixed means is given.

Reverse integral Hardy inequality on metric measure spaces

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. In addition, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.

Applications of convex geometry to Minkowski sums of m ellipsoids in ℝN: Closed-form parametric equations and volume bounds

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of [Formula: see text] arbitrary ellipsoids in [Formula: see text]-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn–Minkowski inequality. A reverse isoperimetric inequality for convex bodies is also given.

The dual Brunn–Minkowski inequality for log-volume of star bodies

This paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and the equivalent Minkowski inequality for mixed log-volume.

Lp -dual three mixed quermassintegrals

In the paper, the concept of Lp -dual three-mixed quermassintegrals is introduced. The formula for the Lp -dual three-mixed quermassintegrals with respect to the p-radial addition is proved. Inequalities of Lp -Minkowski, and Brunn-Minkowski type for the Lp -dual three-mixed quermassintegrals are established. The new Lp -Minkowski inequality is obtained that generalize a family of Minkowski type inequalities. The Lp -Brunn-Minkowski inequality is used to obtain a series of Brunn-Minkowski type inequalities.