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2017-02-14

Duke Math. J. 1993 • Green, M. and Osher, S. Steiner polynomials, Wulff 1. Bonnesen type inequalities. Let K denote a convex body in R2, i.e.

P^ {2}_ {K}- (4\pi-\kappa A_ {K})A_ {K} \geq B_ {K}, (1.6) where B_ {K} vanishes if and only if K is a geodesic disc [ 15, 28 ]. Bonnesen [ 3] established an inequality of the type ( 1.6) in the sphere of radius 1/\sqrt {\kappa}: 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. 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 \(\mathbb{X}_{\kappa}\): We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu's systolic inequality for positively-curved metrics.

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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 2007-08-01 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. BONNESEN-STYLE INEQUALITIES 375 (23) below.

### 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 {\displaystyle L} bounding a domain of area

Below, we shall see that Bonnesen's refinement of the Brunn-Minkowski inequality also follows easily from Wirtinger's inequality. 2016-02-17 Bonnesen's inequality, geometric term; This page lists people with the surname Bonnesen. If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name(s) to the link. This page SOME NEW BONNESEN-STYLE INEQUALITIES 425 Theorem 5. Let D be a plane domain of area A and bounded by a simple closed curve of length L. Let ri and re be, respectively, the radius of the maximum An isoperimetric inequality with applications to curve shortening., Duke Math.

J Korean Math Soc, 2011, 48: 421-430. Google Scholar [36] Zhou J, Du Y, Cheng F. Some Bonnesen-style inequalities for higher dimensions. Acta Math Sin, 2012, 28: 2561-2568.

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Inequality (1) is sharp, since for An isoperimetric inequality with applications to curve shortening., Duke Math.

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. 2007-08-01 · Bonnesen-type inequalities in higher dimensions remain elusive, but perhaps recent generalizations of Hadwiger’s containment theorem to dimensions greater than 2, such as those of Zhou [14,15], may be helpful in develop- ing discriminant inequalities in higher dimension similar to those presented in this article. The venerable isoperimetric inequality, for example, is an easy consequence (see [4]).

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### 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

Let Γ be an oval curve in the Euclidean plane R2 enclosing a domain D of area A. Let P be the length and the curvature of Γ, then 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.

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### 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.

If one let Di · Dj · D, then there is no g 2 G such that gD ‰ D or gD ¾ D.Therefore we have (5) f(A1(D);¢¢¢ ;Al(D)) • 0: This will result in a geometric inequality. (III). Let Di be, respectively, the in-ball and the out-ball of domain Dj (· D), i.e., the largest ball contained in D and the smallest ball containing D. 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.

## 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.

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

It is a strengthening of the classical isoperimetric inequality. The higher-dimensional Bonnesen style inequality in a space of constant curvature is still unknown (cf.