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.

KTH: Isoperometric inequalities and the number of solutions to This result, as well as a sharpening by Bonnesen, can be viewed as a.

a compact convex subset of the plane with non-empty interior. A Bonnesen type inequality is   Others may be found in a recent paper of the author [4] on Bonnesen inequalities and in the book of. Santaló [4] on integral geometry and geometric probability. An   V. Diskant, A generalization of Bonnesen's inequalities, Dokl.

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More precisely, consider a planar simple closed curve of length [math]\displaystyle{ L }[/math] bounding a domain of area [math]\displaystyle{ A }[/math]. For a simple closed curve γ, the stronger inequality due to Bonnesen holds: L 2 − 4 π A ≥ π 2 ( R o u t − R i n) 2 , where, setting Ω = Int ( γ) , R i n and R o u t denote the inner and outer radii of the sets: Bonnesen's inequality: | |Bonnesen's inequality| is an |inequality| relating the length, the area, the radius of t World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. 2018-11-23 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. A standard Bonnesen inequality states that what I call the Bonnesen function (0.1) B(r) = rL - A - nr2 is positive for all r G [rin, r J , where rin , the inradius, is the radius of one of the largest inscribed circles while the outradius rout is the radius of the smallest circumscribed circle. Bonnesen-style inequalities hold true in Rn under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit.

Bonnesen is a surname. Notable people with the surname include: Beatrice Bonnesen, (1906–1979) Danish film actress; Carl Johan Bonnesen, (1868–1933) Danish sculptor; Tommy Bonnesen, (1873–1935) Danish mathematician; See also. Bonnesen's inequality, geometric term

A standard Bonnesen inequality states that what I call the Bonnesen function (0.1) B(r) = rL - A - nr2 is positive for all r G [rin, r J , where rin , the inradius, is the radius of one of the largest inscribed circles while the outradius rout is the radius of the smallest circumscribed circle. Some New Bonnesen-style inequalities.

First, note that we have exhibited nine inequalities of Bonnesen type: (1I)-(13), (16)-(18), and (21)-(23). The last three obviously have all three properties of a Bonnesen inequality, since the right-hand side can vanish only if R = p, in which case the curve must be a circle of radius R. Of the

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.

Zeng, C., Zhou, J., Yue, S.: The symmetric mixed University of Helsinki Faculty of Science Department of Mathematics and Statistics Master’s Thesis 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 Some New Bonnesen-style inequalities. 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. An argument is provided for the equality case of the high dimensional Bonnesen inequality for sections.
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The Bonnesen-style inequality has been extended to surfaces of constant curvature and higher dimensions and many Bonnesen-style inequalities have been found during the past. Mathematicians are still working on Some New Bonnesen-style inequalities. 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.

(2008) [7] and Bonnesen's inequality: | |Bonnesen's inequality| is an |inequality| relating the length, the area, the radius of t World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled.
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An example of this is the Bonnesen inequality for plane figures: $$ F ^ { 2 } - 4 \pi V \geq ( F - 4 \pi r) ^ {2} , $$ where $ r $ is the radius of the largest inscribed circle, and its generalization (see ) for convex bodies in $ \mathbf R ^ {n} $:

Abstract. Abstract In this paper, some Bonnesen-style inequalities on a surface Xκ $\mathbb {X}_{\kappa}$ of constant curvature κ (i.e., the Euclidean plane R2 $\mathbb{R}^{2}$, projective plane RP2 $\mathbb{R}P^{2}$, or hyperbolic plane H2 $\mathbb{H}^{2}$) are proved. 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. 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.