# Bonnesen's inequality is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetric inequality . More precisely, consider a planar simple closed curve of length. L.

We consider the positive centre sets of regular n-gons, rectangles and half discs, and conjecture a Bonnesen type inequality concerning positive centre sets

The area A and the length L of any domain D in the euclidean plane R2 satisfy the inequality (1) L2 ¡4…A ‚ 0: The equality holds if and only if D is a disc. New Bonnesen-type inequalities for simply connected domains on surfaces of constant curvature are proved by using integral formulas. These inequalities are generalizations of known inequalities of The purpose of this paper is to find a new Bonnesen-style inequality with equality condition on surfaces $$\mathbb{X}_{\kappa}$$ of constant curvature, especially on the hyperbolic plane $$\mathbb{H}^{2}$$ by integral geometric method. We are going to seek the following Bonnesen-style inequality for a convex set K in $$\mathbb{X}_{\kappa}$$: The Bonnesen's Inequality states that for a convex plane curve, which has length L and encloses an area A, r L ≥ A + π r 2 for all R in ≤ r ≤ R out where R in is the inradius of the curve, and R out is the circumradius. Bonnesen's inequality for non-simple curves 2 Given a closed curve in the plane R 2, it is well known that L 2 ≥ 4 π A where L is the length of the curve and A is the area of the interior of the curve.

2012-10-01 Bonnesen-style Wulff isoperimetric inequality Zengle Zhang1 and Jiazu Zhou1,2* * Correspondence: [email protected] 1 School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China 2 Southeast Guizhou Vocational College of Technology for Nationalities, Kaili, Guizhou 556000, China Bonnesen type inequality inner parallel body positive centre set regular n-gon MSC classification Primary: 52A10: Convex sets in $2$ dimensions (including convex curves) 2012-05-14 Because of Property 1, any Bonnesen inequality implies the isoperimetric inequality (1). From Property 2, it follows that equality can hold in (1) only when C is a circle. The effect of Property 3 is to give a measure of the curve's "deviation from circularity." Our purpose here is, first, to review what is known for plane domains. In particular, we include ten different inequalities of the In this paper, we obtain some Bonnesen-style Minkowski inequalities of mixed volumes of convex bodies K and L in the Euclidean space Rn. Let L be the unit ball; we get some better Bonnesen-style isoperimetric inequalities than Dinghas’s result for n≥3. Bonnesen's inequality is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetric inequality.

## In this paper we prove a Bonnnesen type inequality for so called s-John domains, s>1, in R^n. We show that the methods that have been applied to John domains in the literature, suitably modified, can be applied to s-John domains. Our result is new and gives a family of Bonnesen type inequalities depending on the parameter s>1.

$\endgroup$ – Jean Marie Aug 8 '16 at 16:18 This page is based on the copyrighted Wikipedia article "Bonnesen%27s_inequality" (); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License.You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. a Bonnesen-type inequality for the sphere, stated in Theorem 2.1. The second main theorem of this article, Theorem 3.1, is a Bonnesen-type inequality for the hyperbolic plane, derived in Section 3.

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When t=\rho _ {m} or Proof. By Blaschke’s rolling theorem (Lemma 2.1 ), we know B_ {t} has no other common point with ∂K when B_ {t} is Theorem 3.2. The The Bonnesen's Inequality states that for a convex plane curve, which has length L and encloses an area A, r L ≥ A + π r 2 for all R in ≤ r ≤ R out where R in is the inradius of the curve, and R out is the circumradius. We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu’s systolic inequality for positively-curved metrics. This page is based on the copyrighted Wikipedia article "Bonnesen%27s_inequality" (); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License.You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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It is a strengthening of the classical isoperimetric inequality. More precisely, consider a planar simple closed curve of length $\displaystyle{ L }$ bounding a domain of area $\displaystyle{ A }$.

Nevertheless, Bonnesen’s inequality holds for arbitrary domains. Bonnesen’s Inequality. The purpose of this paper is to find a new Bonnesen-style inequality with equality condition on surfaces $$\mathbb{X}_{\kappa}$$ of constant curvature, especially on the hyperbolic plane $$\mathbb{H}^{2}$$ by integral geometric method.