Bonnesen-style symmetric mixed inequalities
- Pengfu Wang^{1, 2},
- Miao Luo^{1, 3} and
- Jiazu Zhou^{1}Email author
https://doi.org/10.1186/s13660-016-1146-5
© Wang et al. 2016
Received: 3 January 2016
Accepted: 12 August 2016
Published: 2 September 2016
Abstract
In this paper, we investigate the symmetric mixed isoperimetric deficit \(\Delta_{2}(K_{0},K_{1})\) of domains \(K_{0}\) and \(K_{1}\) in the Euclidean plane \(\mathbb{R}^{2}\). Via the known kinematic formulae of Poincaré and Blaschke in integral geometry, we obtain some Bonnesen-style symmetric mixed inequalities. These new Bonnesen-style symmetric mixed inequalities are known as Bonnesen-style inequalities if one of the domains is a disc. Some inequalities obtained in this paper strengthen the known Bonnesen-style inequalities.
Keywords
isoperimetric inequality symmetric mixed isoperimetric deficit symmetric mixed isoperimetric inequality Bonnesen-style symmetric mixed inequalityMSC
52A10 52A221 Introductions and preliminaries
The classical isoperimetric problem can root back to Ancient Greece. However, a rigorous mathematical proof of the isoperimetric inequality was obtained during the 19th century (see [2, 5–11]). We can find some simplified and beautiful proofs that lead to generalizations of the discrete case, higher dimensions, the surface of constant curvature, and applications to other branches of mathematics [1, 4, 12–34].
Proposition 1.1
Many Bonnesen-style inequalities have been found during the past, and mathematicians are still working on unknown Bonnesen-style inequalities of geometric significance. See [1, 3, 4, 12–16, 19–29, 32, 33, 35–46] for more references.
Containment theorem
Let \(r_{01}^{g}=\max\{t: t(gK_{1})\subseteq K_{0}; g\in G_{2}\}\) and \(R_{01}^{g}=\min\{t: t(gK_{1})\supseteq K_{0}; g\in G_{2}\}\) be the inradius of \(K_{0}\) with respect to \(K_{1}\) and the outradius of \(K_{0}\) with respect to \(K_{1}\), respectively. It is obvious that \(r_{01}^{g}\leq R_{01}^{g}\). Since both \(r_{01}^{g}\) and \(R_{01}^{g}\) are rigid invariant, we simply call them the relative inradius and the relative outradius and denote them \(r_{01}\) and \(R_{01}\), respectively. Note that if \(K_{1}\) is the unit disc, then the relative inradius \(r_{01}\) and the relative outradius \(R_{01}\) become the inscribed radius r and the circumscribed radius R of \(K_{0}\), respectively.
When \(K_{1}\) is the unit disc, then symmetric mixed isoperimetric inequality (1.19) reduces to the isoperimetric inequality (1.9).
The purpose of this paper is to find some new Bonnesen-style symmetric mixed isoperimetric inequalities that strengthen the known Bonnesen-style inequalities.
2 Bonnesen-style symmetric mixed inequality
For any two plane domains \(K_{k}\) (\(k=0,1\)) of areas \(A_{k}\) with simple boundaries of perimeters \(P_{k}\), the convex hulls \(K_{k}^{*}\) of \(K_{k}\) increase the areas \(A_{k}^{*}\) and decrease the perimeters \(P_{k}^{*}\), that is, \(A_{k}^{*}\ge A_{k}\) and \(P_{k}^{*}\le P_{k}\), so that \(P_{0}^{2}P_{1}^{2}-16\pi^{2} A_{0}A_{1}\ge{P_{0}^{*}}^{2}{P_{1}^{*}}^{2}-16\pi^{2} A_{0}^{*}A_{1}^{*}\), that is, \(\Delta_{2}(K_{0},K_{1})\ge\Delta_{2}(K_{0}^{*},K_{1}^{*})\). Therefore, the symmetric mixed isoperimetric inequality and Bonnesen-type symmetric mixed inequality are valid for all domains with simple boundaries in \(\mathbb{R}^{2}\) if these inequalities are valid for convex domains. Hence, from now on, we only consider convex domains when we estimate the lower bounds of the symmetric mixed isoperimetric deficit.
Lemma 2.1
Proof
Remark 2.1
An analogue of inequality (2.1) can already be found in Bol’s work. A complete proof of the analogous inequality (2.1) with equality conditions is given by Böröczky et al. [50] and Luo et al. [51].
When \(K_{1}\) is the unit disc, inequality (2.1) reduces to the following known Bonnesen inequality (see [4, 7, 20, 41]).
Corollary 2.1
Lemma 2.2
Proof
Corollary 2.2
Proof
When \(t=\frac{P_{0}P_{1}}{4\pi A_{1}}\) in (2.16), we immediately obtain (2.17). From the proof of Lemma 2.2 we see that the equality holds in (2.17) if and only if the equalities hold in (2.1) when \(t=R_{01}\) and \(t=\frac{P_{0}P_{1}}{4\pi A_{1}}\), that is, \(R_{01}\) and \(\frac{P_{0}P_{1}}{4\pi A_{1}}\) are roots of the equation \(B_{K_{0},K_{1}}(t)=2\pi A_{1} t^{2}-P_{0}P_{1} t +2\pi A_{0}=0\). It is obvious that \(B_{K_{0},K_{1}}(t)\) reaches the minimum at \(t=\frac{P_{0}P_{1}}{4\pi A_{1}}\), therefore, there is only one root \(R_{01}=\frac{P_{0}P_{1}}{4\pi A_{1}}\) for the equation \(B_{K_{0},K_{1}}(t)=0\), that is, the determinant \(P_{0}^{2}P_{1}^{2}-16\pi^{2}A_{0}A_{1}=0\). By the symmetric mixed isoperimetric inequality (1.19), \(K_{0}\) and \(K_{1}\) are discs. □
Letting \(t=r_{01}\) in inequality (2.16), we immediately obtain the following:
Theorem 2.1
The following Kotlyar inequality (see [17, 40, 47–49]) is an immediate consequence of Theorem 2.1.
Corollary 2.3
When \(t=R_{01}\) in inequality (2.16), we immediately have the following:
Theorem 2.2
We also have the following:
Lemma 2.3
Proof
Corollary 2.4
Letting \(t =R_{01}\) in inequality (2.21), we obtain Theorem 2.1.
When \(t =r_{01}\) in inequality (2.21), we have the following Bonnesen-style symmetric mixed inequality.
Theorem 2.3
The lower bound of symmetric mixed isoperimetric deficit in inequality (2.18) or (2.20) is the maximum of the function \(f(t)\). The lower bound of symmetric mixed isoperimetric deficit in inequality (2.18) or (2.23) is the maximum of the function \(g(t)\). Which one is the best lower bound of symmetric mixed isoperimetric deficit in inequalities (2.18), (2.20), and (2.23)?
When \(K_{1}\) is the unit disc, these Bonnesen-style symmetric mixed inequalities immediately lead to the following known Bonnesen-style inequalities of Burago, Grinberg, Hsiung, Hadwiger, Osserman, Zhou, and Ren (see [9, 12, 16, 20, 21, 41, 44]).
Corollary 2.5
Declarations
Acknowledgements
The corresponding author is supported in part by the NSFC (No. 11271302) and the PhD Program of Higher Education Research Fund (No. 2012182110020). The first author and the second author are supported in part by NSFC (No. 11401486).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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