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. The limiting case as κ → 0 in either of Theorems 2.1 and 3.3 yields the classical Bonnesen inequality (1), as described above.

Abstract. An argument is provided for the equality case of the high dimensional Bonnesen inequality for sections. The known equality case of the Bonnesen inequality for …

1 Introduction Perhaps the oldest geometric inequality is the following isoperimetric inequality: Theorem 1. 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.

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. 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 .

An argument is provided for the equality case of the high dimensional Bonnesen inequality for sections. The known equality case of the Bonnesen inequality for projections is presented as a consequence.
An isoperimetric inequality with applications to curve shortening., Duke Math.

ABSTRACT. Two Bonnesen-style inequalities are obtained for the relative in-radius of one convex body with respect to another in n-dimensional space. Both reduce to the known planar inequality; one sharpens the relative isoperi-metric inequality, the other states that a quadratic polynomial is negative at the inradius.

Let K denote a convex body in R2, i.e. a compact convex subset of the plane with non-empty interior.

Zeng, C., Ma, L., Zhou, J., Chen, F.: The Bonnesen isoperimetric inequality in a surface of constant curvature. Zeng, C., Zhou, J., Yue, S.: The symmetric mixed

Many Bonnesen-style inequalities have been found (see [ 1, 4, 12, 16, 19, 33 ]). Conversely, we considered the upper bound of the isoperimetric deficit, that is, \Delta _ {2} (K)=P^ {2}-4\pi A\le U_ {K}, In this paper, we establish some Bonnesen-style Wulff isoperimetric inequalities and reverse Bonnesen-style Wulff isoperimetric inequalities. Those inequalities obtained are extensions of known Bonnesen-style inequalities and reverse Bonnesen-style inequalities. Introduction and main results 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 bounding a domain of area . a Bonnesen-type inequality for the sphere, stated in Theorem 2.1.

References. [1] T. Bonnesen, "Ueber eine Verschärferung der isoperimetische Ungleichheit des Kreises in der Ebene und auf die 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. The limiting case as κ → 0 in either of Theorems 2.1 and 3.3 yields the classical Bonnesen inequality (1), as described above. Bonnesen’s inequality for non-convex sets by using the convex hull is that unlike the circumradius, which is the same for the convex hull and for the original domain, the inradius of the convex hull may be larger that that of the original domain. Nevertheless, Bonnesen’s inequality holds for arbitrary domains. Bonnesen’s Inequality.
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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.

Theorem. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant Gauss curvature κ > 0) and the hyperbolic plane Seminar on Differential Geometry. (AM-102), Volume 102.
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known spherical/hyperbolic isoperimetric inequality, allows to solve the isodiametric equalities (e.g. a spherical Bonnesen-type isodiametric inequality for cen-.

Below, we shall see that Bonnesen's refinement of the Brunn-Minkowski inequality also follows easily from Wirtinger's inequality. On Bonnesen-type isoperimetric inequalities 59 (II).

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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.

The Bonnesen inequality (see [1,2]) is A Ls + ˇs2 0; s 2[r in;r out]: (1.4) Using this and symmetrisation, Gage [4] successfully proved an inequality for the 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. Zeng, C., Ma, L., Zhou, J., Chen, F.: The Bonnesen isoperimetric inequality in a surface of constant curvature.