�ber die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfl�che und in der Ebene

Bernstein, Felix

doi:10.1007/bf01447496

Cited by 63 publications

(33 citation statements)

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“…One way to prove that almost isoperimetric domains are close to minimizers is to prove that they contain a minimizer of radius r and are included in another of radius R, with small difference of radii R − r and the same center. In euclidean space, such inclusions imply that the domain under consideration is at Hausdorff distance at most (R − r)/2 from some ball; the Bonnesen inequality gives a bound on the possible values of R − r in terms of the isoperimetric deficit, and implies results similar to ours in the euclidean case; see [Ber05,Bon29,Oss78]. However, balls and minimizers are different in the L 1 plane, so that if A is between concentric squares of radii R and r, one can only say that it is at L 1 Hausdorff distance R − r from some square, while the bound on the L ∞ Hausdorff distance is the expected (R − r)/2.…”

Section: Statement Of the Resultssupporting

confidence: 54%

Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

Kloeckner

2010

Proc. Amer. Math. Soc.

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Abstract. An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be.The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the L 1 case has drawn much less attention.In this note we prove two quantitative isoperimetric inequalities in the L 1 Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch. Statement of the resultsWe consider the plane R 2 endowed with the L 1 metricWe denote the boundary of a set by ∂. The notation | · | shall be used to denote the size of an object, whatever its nature. If A is a measurable plane set, then |A| is its Lebesgue measure, also called its area; if v is a vector, |v| is its L 1 norm; if γ is a curve, |γ| is its L 1 length, namelywhere the supremum is over all increasing sequences (t 1 , t 2 , . . . , t n ) taking values in the definition interval of γ. A curve is said to be (L 1 -)rectifiable if its length is finite. For example, a curve defined on a segment whose coordinate functions are both monotonic is rectifiable, and its length is the distance between its endpoints. See e.g. [AT04] for more information on the length of curves in a metric space.By a domain of the plane, we mean the closure of the bounded component of a Jordan curve. In particular, domains are compact and connected. All rectangles and squares considered are assumed to have their sides parallel to the coordinate

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Section: Statement Of the Resultssupporting

confidence: 54%

Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

Kloeckner

2010

Proc. Amer. Math. Soc.

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“…Bernstein [7] extended the classical isoperimetric inequality to domains on S 2 . The generalization to other spaces of constant curvature is due to Schmidt [20].…”

Section: Inequalities For the Distribution Function Of The Capacity Pmentioning

confidence: 99%

Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature

Bandle¹,

Brillard

Flucher³

1998

Trans. Amer. Math. Soc.

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Abstract. We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding LaplaceBeltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.

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“…Inequalities of this kind have been named by Osserman [19] Bonnesen type inequalities, following the results proved in the plane by Bonnesen in 1924 (see [4] and also [2]). More precisely, Osserman calls in this way any inequality of the form λ(E) ≤ P (E) 2 − 4π|E| , valid for smooth sets E in the plane R 2 , where the quantity λ(E) has the following three properties: (i) λ(E) is nonnegative; (ii) λ(E) vanishes only when E is a ball; (iii) λ(E) is a suitable measure of the "asymmetry" of E.…”

Section: Introductionmentioning

confidence: 99%