The Quantitative Isoperimetric Inequality for Planar Convex Domains

Carlo Nitsch

Displaying similar documents to “The Quantitative Isoperimetric Inequality for Planar Convex Domains”

A sharp isoperimetric inequality in the plane

Angelo Alvino, Vincenzo Ferone, Carlo Nitsch (2011)

Journal of the European Mathematical Society

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We show that among all the convex bounded domain in $m a t h b b R^{2}$ having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.

Weak lineal convexity

Christer O. Kiselman (2015)

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A bounded open set with boundary of class C¹ which is locally weakly lineally convex is weakly lineally convex, but, as shown by Yuriĭ Zelinskiĭ, this is not true for unbounded domains. The purpose here is to construct explicit examples, Hartogs domains, showing this. Their boundary can have regularity $C^{1, 1}$ or $C^{\infty}$ . Obstructions to constructing smoothly bounded domains with certain homogeneity properties will be discussed.

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Fethi Kadhi (2002)

RAIRO - Operations Research - Recherche Opérationnelle

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We investigate the minima of functionals of the form $\int_{[a, b]} g (\dot{u} (s)) d s$ where $g$ is strictly convex. The admissible functions $u : [a, b] \to ℝ$ are not necessarily convex and satisfy $u \leq f$ on $[a, b]$ , $u (a) = f (a)$ , $u (b) = f (b)$ , $f$ is a fixed function on $[a, b]$ . We show that the minimum is attained by $\bar{f}$ , the convex envelope of $f$ .

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M. Obradović, S. Owa (1986)

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Poincaré Inequalities and Moment Maps

Bo’az Klartag (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in $ℝ^{n}$ . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of $ℓ_{p}$ -spaces in $ℝ^{n}$ for $0 < p < 1$ .

The Young inequality and the Δ₂-condition

Philippe Laurençot (2002)

Colloquium Mathematicae

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If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality $x y \leq ε φ (x) + C_{ε} φ * (y)$ is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.

Balls for the Kobayashi distance and extension of the automorphisms of strictly convex domains in $C^{n}$ with real analytic boundary

Andrea Iannuzzi (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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It is shown that given a bounded strictly convex domain $Ω$ in $C^{n}$ with real analitic boundary and a point $x_{0}$ in $Ω$ , there exists a larger bounded strictly convex domain $Ω^{'}$ with real analitic boundary, close as wished to $Ω$ , such that $Ω$ is a ball for the Kobayashi distance of $Ω^{'}$ with center $x_{0}$ . The result is applied to prove that if $Ω$ is not biholomorphic to the ball then any automorphism of $Ω$ extends to an automorphism of $Ω^{'}$ .

On the algebra of $A^{k}$ -functions

Ulf Backlund, Anders Fällström (2006)

Mathematica Bohemica

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For a domain $Ω \subset ℂ^{n}$ let $H (Ω)$ be the holomorphic functions on $Ω$ and for any $k \in ℕ$ let $A^{k} (Ω) = H (Ω) \cap C^{k} (\bar{Ω})$ . Denote by $𝒜_{D}^{k} (Ω)$ the set of functions $f Ω \to [0, \infty)$ with the property that there exists a sequence of functions $f_{j} \in A^{k} (Ω)$ such that ${| f_{j} |}$ is a nonincreasing sequence and such that $f (z) = {lim}_{j \to \infty} | f_{j} (z) |$ . By $𝒜_{I}^{k} (Ω)$ denote the set of functions $f Ω \to (0, \infty)$ with the property that there exists a sequence of functions $f_{j} \in A^{k} (Ω)$ such that ${| f_{j} |}$ is a nondecreasing sequence and such that $f (z) = {lim}_{j \to \infty} | f_{j} (z) |$ . Let $k \in ℕ$ and let $Ω_{1}$ and $Ω_{2}$ be bounded $A^{k}$ -domains of holomorphy in $ℂ^{m_{1}}$ and $ℂ^{m_{2}}$ respectively. Let $g_{1} \in 𝒜_{D}^{k} (Ω_{1})$ , $g_{2} \in 𝒜_{I}^{k} (Ω_{1})$ and $h \in 𝒜_{D}^{k} (Ω_{2}) \cap 𝒜_{I}^{k} (Ω_{2})$ . We prove...

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations $D u \in K$ , where $u : Ω \subset ℝ^{n} \to ℝ^{m}$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of $D u$ and second we replace Gromov’s $P$ −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

Canonical mappings in $Σ_{M} (D)$

Z. Abdul-Hadi, D. Bschouty, W. Hengartner (1985)

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Smoothing a polyhedral convex function via cumulant transformation and homogenization

Alberto Seeger (1997)

Annales Polonici Mathematici

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Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family ${g ₜ}_{t > 0}$ which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family ${g ₜ}_{t > 0}$ involves the concept of cumulant transformation and a standard homogenization procedure.

Convex universal fixers

Magdalena Lemańska, Rita Zuazua (2012)

Discussiones Mathematicae Graph Theory

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In [1] Burger and Mynhardt introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G’ a copy of G. For a bijective function π: V(G) → V(G’), define the prism πG of G as follows: V(πG) = V(G) ∪ V(G’) and $E (π G) = E (G) \cup E (G^{'}) \cup M_{π}$ , where $M_{π} = u π (u) | u \in V (G)$ . Let γ(G) be the domination number of G. If γ(πG) = γ(G) for any bijective function π, then G is called a universal fixer. In [9] it is conjectured that the only universal fixers are the edgeless graphs K̅ₙ. In this work we generalize the concept...