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Register a new userOn The Quantitative Isoperimetric Inequality In The Plane
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In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $Omega$, different from a ball, which minimizes the ratio $delta(Omega)/lambda^2(Omega)$, where $delta$ is the isoperimetric deficit and $lambda$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.
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We prove a counterpart of the log-convex density conjecture in the hyperbolic plane.
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. In the first case, we determine the extremal domains.
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 $mathbb{R}^2$ is written $P_f(E)$. We study minimisers for the weighted isoperimetric problem [ I_f(v):=infBig{ P_f(E):Etext{ is a set of finite perimeter in }mathbb{R}^2text{ and }V_f(E)=vBig} ] for $v>0$. Suppose $f$ takes the form $f:mathbb{R}^2rightarrow(0,+infty);xmapsto e^{h(|x|)}$ where $h:[0,+infty)rightarrowmathbb{R}$ is a non-decreasing convex function. Let $v>0$ and $B$ a centred ball in $mathbb{R}^2$ with $V_f(B)=v$. We show that $B$ is a minimiser for the above variational problem and obtain a uniqueness result.
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 inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromovs proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J. Hersch, 1970), k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) and k=3 (N. Nadirashvili and Y. Sire, 2017). In particular, we argue that for any k>=2, the supremum of the k-th nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannian metrics which are smooth outsitde a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres, the key new ingredient being a bound on the harmonic degree of a harmonic map into a sphere obtained by N. Ejiri.
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