A reverse isoperimetric inequality for embedded starshaped plane curves

In this note, we present a reverse isoperimetric inequality for embedded starshaped closed plane curves, which states that if K is a starshaped domain with perimeter p(K) and area a(K), then one gets

$$\begin{aligned} p(K)^2 \le 4\pi \Big ((a(K)+\tilde{a}(K)\Big ), \end{aligned}$$

where \(\tilde{a}(K)\) denotes the oriented area of the domain enclosed by \(\beta \) (defined in Section 2), and equality holds if and only if K is a disc.

     

The reverse isoperimetric inequality for convex plane curves through a length-preserving flow

Archiv der Mathematik - By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338,...     

Stability of a reverse isoperimetric inequality

In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303-308], which states that if K is a convex domain in the plane with perimeter p(K) and area a(K), then one gets , where denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc.     

Relative frames on J-holomorphic curves

The aim of this work is to prove that any non-constant J-holomorphic disc with its boundary in a given Lagrangian submanifold can be decomposed in homology into a sum of finitely many J-holomorphic simple discs with the same Lagrangian boundary condition. As a consequence, in dimension higher than 6, any generic J-holomorphic disc is multicovered.     

A quantitative isoperimetric inequality for fractional perimeters

Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.     

Bonnesen's inequality for the isoperimetric deficiency of closed curves in the plane

An inequality of T. Bonnesen for the isoperimetric deficiency of a convex closed curve in the plane is extended to arbitrary simple closed curves. As a primary tool it is shown that, for any such curve, there exist two concentric circles such that the curve is between these and passes at least four times between them. Next, it is shown that Bonnesen's inequality holds for any such pair of concentric circles.This work was supported by The Danish Natural Science Council and carried out during the author's stay 1989–90 as a Visiting Member at the Institute for Advanced Study, Princeton, N.J., U.S.A.     

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

A sharp isoperimetric inequality for rank one symmetric spaces

 We prove an isoperimetric inequality for compact, regular domains in rank one symmetric spaces, which is sharp for geodesic balls. Besides volume and area of a given domain, some weak information about the second fundamental form of its boundary is involved. Received: 2 September 2002 / Revised version: 10 December 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 53C35, 52A40, 51M25     

An isoperimetric inequality for the Wiener sausage

Let (ξ(s)) _{s?≥ 0} be a standard Brownian motion in d?≥ 1 dimensions and let (D _s ) _{s ≥?0} be a collection of open sets in ${\mathbb{R}^d}$ . For each s, let B _s be a ball centered at 0 with vol(B _s ) =?vol(D _s ). We show that ${\mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + D_s))] \geq \mathbb{E}[\rm {vol}(\cup_{s \leq t}(\xi(s) + B_s))]}$ , for all t. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.     

An isoperimetric inequality for the Heisenberg groups

We show that the Heisenberg groups of dimension five and higher, considered as Riemannian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area ~ L ².) This implies several important results about isoperimetric inequalities for discrete groups that act either on or on complex hyperbolic space, and provides interesting examples in geometric group theory. The proof consists of explicit construction of a disk spanning each loop in . Submitted: April 1997, Final version: November 1997     

On the linear isoperimetric inequality

This paper contains a sharp version of the well-known linear isoperimetric inequality for minimal surfacesX area(X)1/2oscillation(X)length(X).Supported by Sonderforschungsbereich 72 der Deutschen Forschungsgemeinschaft at Bonn University.     

A strong form of the quantitative isoperimetric inequality

We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary.