Weighted quantitative isoperimetric inequalities in the Grushin space R h + 1 ${R}^{h+1}$ with density | x | p $|x|^{p}$

The Isoperimetric Problem in the Grushin Space R With Density |x|

In this paper we study the isoperimetric problem in a class of x-spherically symmetric sets in the Grushin space R with density |x|, p > −h + 1. First we prove the existence of weighted isoperimetric sets. Then we deduce that, up to a vertical translation, a dilation and a negligible set, the weighted isoperimetric set is only of the form { (x, y) ∈ R : |y| < ∫ π 2 arcsin |x| sin (t)dt, |x| < 1...

Quantitative Isoperimetric Inequalities in H

In the Heisenberg group H, n ≥ 1, we prove quantitative isoperimetric inequalities for Pansu’s spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.

1. ISOPERIMETRIC AND CONCENTRATION INEQUALITIES p. 7

Weighted quantitative isoperimetric inequalities in the Grushin space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${R}^{h+1}$\end{document}Rh+1 with density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|x|^{p}$\end{document}|x|p

*Correspondence: [email protected] 1School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China 2School of Mathematics and Computer Science, Anhui Normal University, Wuhu, 241000, China Abstract In this paper, we prove weighted quantitative isoperimetric inequalities for the set Eα = {(x, y) ∈ Rh+1 : |y| < ∫ π 2 arcsin |x| sin α+1(t)dt, |x| < 1} in half-cylind...

On weighted isoperimetric and Poincaré-type inequalities

Weighted isoperimetric and Poincaré-type inequalities are studied for κ-concave probability measures (in the hierarchy of convex measures).