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.     

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.     

An isoperimetric inequality related to a Bernoulli problem

Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in Flucher et al. (Reine Angew Math 486:165–204, 1997).     

An isoperimetric inequality in the plane with a log-convex density

Ricerche di Matematica - 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...     

An isoperimetric inequality in the universal cover of the punctured plane

We find the largest ? (approximately 1.71579) for which any simple closed path α in the universal cover of R²?Z², equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥?A(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R²?Z².     

A note on the isoperimetric inequality

We show that the sharp integral form on the isoperimetric inequality holds for those orientation-preserving mappings whose Jacobians obey the rule of integration by parts.

     

Perturbations in the Gaussian isoperimetric inequality

Concentration and logarithmic Sobolev inequalities are derived for a class of multidimensional probability distributions, including spherically invariant log-concave measures. Bibliography: 17 titles.     

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.     

An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift

We generalize the classical Rayleigh–Faber–Krahn inequality to the case of the Dirichlet Laplacian with a drift. We also solve some optimization problems for the principal eigenvalue of the operator $?\mathrm{\Delta }+v?\mathrm{?}$ in a fixed domain with a control of the drift v in $L^{\infty }$. To cite this article: F. Hamel et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Petty's projection inequality and Santalo's affine isoperimetric inequality

Using results of K. Kiener and the Riesz-Sobolev convolution inequality we give a new proof of Petty's projection inequality. By the same method we also obtain a proof of Santalo's affine isoperimetric inequality.Supported in part by BSF and Erwin Schrödinger Auslandsstipendium J0630, J0804.     

An isoperimetric inequality in the optimization of circular rings under normal pressure

The problem of determining the optimal cross section of a circular ring so as to maximize the buckling pressure under a given total volume is formulated and solved. An isoperimetric inequality is proved: Among all the circular rings of given mass and radius, the ring with constant bending rigidity along the arc length has the largest critical buckling pressure.     

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.     

An isoperimetric inequality for the first eigenfunction in the fixed membrane problem

Summary An isoperimetric inequality is obtained which relates theL ₁-andL ₂-integrals of the first eigenfunction in the problem of the vibrating clamped membrane.

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Zusammenfassung Für das Problem der eingespannten schwingenden Membran wird zwischen demL ₁-und demL ₂-Integral der ersten Eigenfunktion eine isoperimetrische Ungleichung hergeleitet.