2012-05-14

Fenchel , Werner ; Bonnesen, Tommy (1934). Theorie der konvexen Körper . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 . Berlin: 1.

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. The limiting case as κ → 0 in either of Theorems 2.1 and 3.3 yields the classical Bonnesen inequality (1), as described above. A Bonnesen-type inequality in \mathbb {X}_ {\kappa} is of the form. 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 is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve.

Bonnesen inequality

We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu's systolic inequality for positively-curved metrics. The remainder term in the inequality, analogous to that in Bonnesen's inequality, is a function of R-r (suitably normalized), where R and r are respectively the circumradius and the inradius of the Weyl-Lewy Euclidean embedding of the orientable double cover. 1. Bonnesen type inequalities. Let K denote a convex body in R2, i.e. a compact convex subset of the plane with non-empty interior.

Bonnesen type inequalities. Let K denote a convex body in R2, i.e.

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.

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. Bonnesen [2], Bonnesen and Fenchel [3], Schneider [9] and the survey by Osserman [6], which is an excellent guide in the world of these inequalities.

On Bonnesen-style symmetric mixed inequality of two planar convex domains. PF WANG, WX XU, JZ ZHOU, BC ZHU. SCIENTIA SINICA Mathematica 45 (3),

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 (1987).

References. [1] T. Bonnesen, "Ueber eine Verschärferung der isoperimetische Ungleichheit des Kreises in der Ebene und auf die First, note that we have exhibited nine inequalities of Bonnesen type: (1I)-(13), (16)-(18), and (21)-(23).
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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. 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.

KTH: Isoperometric inequalities and the number of solutions to This result, as well as a sharpening by Bonnesen, can be viewed as a. Först ska "Inequality regimes" av Joan Acker diskuteras.
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A sharp reverse Bonnesen-style inequality and generalization. Abstract. We investigate the isoperimetric deficit of the oval domain in the Euclidean plane.

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.

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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 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. Bonnesen’s inequality will relate the circumradius, inradius, A and L. One of the complications in proving 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 In this paper, some Bonnesen-style inequalities on a surface X κ of constant curvature κ (i.e., the Euclidean plane R 2, projective plane R P 2, or hyperbolic plane H 2) are proved. The method is integral geometric and gives a uniform proof of some Bonnesen-style inequalities alone with equality conditions. domain to contain another and Bonnesen-type isoperimetric inequalities. 1 Introduction Perhaps the oldest geometric inequality is the following isoperimetric inequality: Theorem 1.