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We consider the punctured plane with volume density $|x|^alpha$ and perimeter density $|x|^beta$. We show that centred balls are uniquely isoperimetric for indices $(alpha,beta)$ which satisfy the conditions $alpha-beta+1>0$, $alphaleq 2beta$ and $alpha(beta+1)leqbeta^2$ except in the case $alpha=beta=0$ which corresponds to the classical isoperimetric inequality.
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequali
Given a positive lower semi-continuous density $f$ on $mathbb{R}^2$ the weighted volume $V_f:=fmathscr{L}^2$ is defined on the $mathscr{L}^2$-measurable sets in $mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$ in $mathbb{R}^
We prove that a plane domain which is almost isoperimetric (with respect to the $L^1$ metric) is close to a square whose sides are parallel to the coordinates axis. Closeness is measured either by $L^infty$ Haussdorf distance or Fraenkel asymmetry. I
We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $tau$ functions. By using
In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderons and Zygmunds theory of first order differentiability for functi