THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATURE

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1 J Korean Math Soc ), No 6, pp THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATURE Jiazu Zhou and Fangwei Chen Reprinted from the Journal of the Korean Mathematical Society Vol 44, No 6, November 007 c 007 The Korean Mathematical Society

2 J Korean Math Soc ), No 6, pp THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATURE Jiazu Zhou and Fangwei Chen Abstract We investigate the containment measure of one domain to contain in another domain in a plane X κ of constant curvature We obtain some Bonnesen-type inequalities involving the area, length, radius of the inscribed and the circumscribed disc of a domain D in X κ 1 Introduction A geometric inequality describes the relation between invariants of geometric subjects Perhaps the best and the most remarkable one is the classical isoperimetric inequality that relates volume to area of a plane domain: Among domains with fixed areas the disc has the shortest circumlength That is, the domain D with area A and length L satisfies L 4πA 0, with equality if and only if D is a disc The isoperimetric inequality has been generalized to higher dimensions, which has been the object of much research in the last century, still going on today Its applications reach algebra, differential geometry, differential equations and many mathematical areas One can find the literature from references 1], 4], 5], 6] The following inequalities are known see 1], 4], 5], 9]) Proposition 1 Let D be a domain of area A and bonded by a simple closed curve of length L in the Euclidean plane R Let r i and r e be, respectively, the radius of the inscribed disc and the circumscribed disc Then for any disc of Received March 1, 006 Revised October 17, Mathematics Subject Classification Primary 5A, 53C65 Secondary 51C16 Key words and phrases isoperimetric inequality, Bonessen inequality, kinematic measure, containment measure, hyperbolic plane, projective plane, geodesic disc Supported in part by Chinese NSF grant number: ) 1363 c 007 The Korean Mathematical Society

3 1364 JIAZU ZHOU AND FANGWEI CHEN radius r r i r r e ), we have the following inequalities: 11) L 4πA 0 L 4πA π r e r i πr Lr + A 0 L 4πA L πr L 4πA ) L A r L 4πA A r πr L 4πA A 1 r i 1 r e L 4πA L r e r i r e +r i L 4πA A 1 1 r L 4πA L r ri r+r i r i L 4πA A 1 r 1 r e L L 4πA π r i r r e L+ L 4πA L 4πA L re r r e +r π Any one equality of above holds when and only when D is a disc The second inequality of 11) is called the Bonnesen isoperimetric inequality We define the isoperimetric deficit of D as D) L 4πA Then we can see the geometric meaning of inequalities 11) D) measures the deficit between a domain D and the disc One hope to obtain the Bonnesen-type inequalities for domains in the higher dimension spaces Refer to 3, 7, 8, 9, 10, 11, 1] for more results about the containment measures In this paper, we hope to obtain the Bonnesen-type inequalities for domains in a plane X κ of constant curvature κ The methods could result isoperimetric inequalities for higher dimensions Let D k k = i, j) be a domain in the ambient space X κ, the plane of constant curvature κ Thus X κ is either the Euclidean plane R κ = 0), the projective plane RP κ > 0), or the hyperbolic plane H κ < 0) We assume that D k is a rectifiable simple closed curve The area and perimeter length of D k is denoted by A k and L k, respectively, or simply A and L Let G κ be the group of isometry in X κ and dg be the kinematic measure Haar measure in measure theory) on G κ We consider the following containment measure m{g G κ : gd j D i or gd j D i } = dg {g G 1) κ:gd j D i or gd j D i} = dg dg {g G κ:d i gdj } {g G κ: D i g Dj } If we can estimate the last integral from above and the integral dg {g G κ:d i gdj } from below in terms of geometric invariants of D k, then we obtain an inequality of the form 13) m{g G κ : gd j D i or gd j D i } fi 1 i,, I l i I 1 j,, I l j), where each of I α k k = i, j 1 α l) is an integral geometric invariant of D k

4 THE BONNESEN-TYPE INEQUALITIES IN A PLANE 1365 One can then immediately state the following conclusions: 1 If fi 1 i,, Il i I1 j,, Il j ) > 0 then there is an isometry g G κ such that either gd j contains or is contained in D i If one let D i D j D), then there is no g G κ such that gd D or gd D Hence we have 14) fi 1 D),, I l D)) 0 This is an geometric inequality of domain D 3 Let D i be, respectively, the in-disc and the out-disc of domain D j D), that is, the largest inscribed disc contained in D and the smallest circumscribed disc containing D Then there is no g G κ such that gd D i or gd D i Therefore we have 15) fi 1 D),, I l D), r e ) 0, fi 1 D),, I l D), r i ) 0, where r e and r i are, respectively, the circumscribed radius and inscribed radius of D From these inequalities one will obtain the Bonnesen inequality the second inequality in 11)) in a plane X κ of constant curvature 4 If one let D i be a disc of radius r between the inscribed disc of radius r i and the circumscribed disc of radius r e of D j D) Then repeating the same procedure of above will lead to following inequality 16) fi 1 D),, I l D), r) 0 i r r e It is usually called the Bonnesen-type inequality Above ideas is due to the first author see 9, 10, 11, 1, 13, 14, 15]) and he obtain some Bonnesen-type inequalities in Proposition 1 for domain D in the Euclidean plane In this paper, we follow Zhou s idea and use the containment measure of Grinbeg, Ren and Zhou see ]) for a plane X κ of constant curvature κ We obtain some Bonnesen-type inequalities for domains in either a hyperbolic plane or a projective plane Zhou s idea could result more Bonnesen-type inequalities for higher dimensions if appropriate containment measure of domains are achieved see 10, 11, 1, 13, 14, 15]) Bonnesen-type inequalities Let D k k = i, j) be domains in a plane X κ of constant curvature κ For g G κ the group of isometry of X κ Grinberg, Ren and Zhou have the following containment measure inequality see ]): 1) m{g G κ : gd j D i or gd j D i } = dg πa i + A j ) L i L j κa i A j {g G κ:gd j D i or gd j D i} If we let D i D j D, then there is no g G κ such that gd D or gd D and the containment measure inequality 1) immediately result in

5 1366 JIAZU ZHOU AND FANGWEI CHEN the following isoperimetric inequality in X κ ) L 4πA + κa 0 For a disc of radius r in the hyperbolic plane H, that is, κ = 1, we have 3) L = π sinh r, A = 4π sinh r Therefore let D i = D and let D j be a disc of radius r between the inscribed disc of radius r i and the circumscribed disc of radius r e of D We have neither gd j D nor gd j D for any g G κ Then the measure m{g G κ : gd j D or gd j D} = 0 and the inequality 1) leads to 4) L sinh r 4π + A) sinh r A 0, r i r r e ) Using the equalities 5) sinh x = sinh x cosh x, 1 tanh 1 x = cosh x and the formula 4) we have 6L tanh r A 4π + A) tanh r 0 Letting ψr) = L tanh r A 4π + A) tanh r immediately gives 7) L 44π + A) A = 4π + A) r i r r e ) L 4π + A) tanh r ] + ψr) In special cases when r = r i and r e, respectively, the equality 7) also hold, that is, L 44π +A) A = 4π L + A) 4π +A) tanh r i ] + ψri ), 8) L 44π +A) A = 4π L + A) 4π +A) tanh r e ] + ψre ) Since ψr) 0 r i r r e ), we have L 44π +A) A 4π + A) 9) L 44π +A) A 4π + A) L 4π +A) tanh re By adding two inequalities of 9) we have 10) L 44π + A) A 4π + A) 4 tanh ri ], L 4π +A)] tanh r e tanh r i,

6 THE BONNESEN-TYPE INEQUALITIES IN A PLANE 1367 that is, 11) L 4πA A 4π + A 4 tanh r e tanh r i We proved the following: Theorem 1 Let D be a domain of area A and bonded by a simple closed curve of length L in the hyperbolic plane H Let r i and r e be, respectively, the radius of the inscribed disc and the circumscribed disc Then for any disc of radius r r i r r e ), we have the following inequalities: 1) L 4πA A 0 L 4πA A 4π +A 4π + A) tanh r 4 tanh re L tanh r + A 0 tanh ri The second inequality of 1) can be rewritten in several equivalent forms: Theorem Let D be a domain of area A and bonded by a simple closed curve of length L in the hyperbolic plane H Let r i and r e be, respectively, the radius of the inscribed disc and the circumscribed disc Then for any disc of radius r r i r r e ), we have the following inequalities: 13) L 4πA A L 4πA A L 4πA A L 4π +A tanh r A L tanh r A tanh r From the second formula of 13) we have 14) L 4πA A L A tanh re Adding inequalities 14) yields 15) L 4πA A A 1 4 tanh r i π + A ) tanh r ] L 4πA A A tanh ri 1 tanh r e L and 1 tanh r e, respec- 1 Adding inequalities 14) after multiplied by tanh r i tively, gives 16) L 4πA A L tanh r e tanh r i tanh re + tanh ri

7 1368 JIAZU ZHOU AND FANGWEI CHEN Notice that the equation L tanh r A 4π + A) tanh roots L 4πA A 17) tanh r k = L ± So we obtain 18) L L 4πA A 4π + A and we proved the following 4π + A k = i, e r = 0 has two tanh r i tanh r L + L 4πA A e 4π, + A Theorem 3 Let D be a domain of area A and bonded by a simple closed curve of length L in the hyperbolic plane H Let r i and r e be, respectively, the radius of the inscribed disc and the circumscribed disc Then we have 19) L 4πA A A 4 L 4πA A L L L 4πA A 4π +A 1 tanh r i 1 tanh r e tanh r e tanh r i tanh r e +tanh r i tanh ri re tanh L+ Each equality holds when and only when D is a disc L 4πA A 4π +A The equation L tanh r A 4π + A) tanh r = 0 has an unique root when and only when L 4πA A = 0 This leads to tanh r i = tanh r i We conclude that each equality of those inequalities in Theorem 1, Theorem and Theorem 3 holds when and only when D is a geodesic disc In the case of that D is a domain in the projective plane P R, that is, κ = 1 For a geodesic disc of radio r we have 0) L = π sin r ), A = 4π sin r ), where r π ) We use the same method we just used in hyperbolic plane, let D i D and let D j be a disc of radius r between the inscribed disc of radius r i and the circumscribed disc of radius r e of D where we assume that the r e π ) We have neither gd j D nor gd j D for any g G κ Then the measure m{g G κ : gd j D or gd j D} = 0 and the inequality 1) leads to 1L tan r A 4π A) tan r 0 Let ) φr) = L tan r A 4π A) tan r 0

8 THE BONNESEN-TYPE INEQUALITIES IN A PLANE 1369 Then we obtain L 44π A) A = 4π A) L 4π A) tan r ] + φr) In special cases when r = r i and r e, respectively, the equality ) also hold, that is, L 44π A) A = 4π L ri A) 4π A) tan ] + φri ), 3) L 44π A) A = 4π L re A) 4π A) tan ] + φre ) Since φr) 0 r i r r e ), we have L 44π A) A 4π A) 4) L 44π A) A 4π A) L 4π A) tan re By adding two inequalities of 4) we have 5) that is, L 44π A) A 4π A) 4 6) L 4πA + A 4π A 4 tan ri ], L 4π A)] tan r e tan r i, tan r e tan r i If we let D i D j D, then the containment measure inequality for the case of projective plan P R gives 7) L 4πA + A 0 Since has two roots for tan r : φr) = L tan r A 4π A) tan r = 0, 8) tan r k = L ± therefore we have 9) L L 4πA + A 4π A L 4πA + A 4π A k = i, e, tan r i tan r L + L 4πA + A e 4π A The equation φr) = 0 has a unique root when and only when L 4πA+ A = 0 hence tan r i = tan r e This means that D is domain bounded by a geodesic circle D

9 1370 JIAZU ZHOU AND FANGWEI CHEN We proved the following Theorem 4 Let D be a domain of area A and bonded by a simple closed curve of length L in the projective plane P R Let r i and r e r e π ) be, respectively, radius of the inscribed disc and the circumscribed disc Then for any disc of radius r r i r r e ), we have the following inequalities: 30) L 4πA + A 0 L 4πA + A 4π A A L tan r L L 4πA+ A 4π A tan r e tan r i + 4π A) tan r 0 4 tan ri re tan L+ Each equality holds when and only when D is a geodesic disc L 4πA+ A 4π A The second inequality of 30) in Theorem 4 can be rewritten in several equivalent forms, that is: Theorem 5 Let D be a domain of area A and bonded by a simple closed curve of length L in the projective plane P R Let r i and r e r e π ) be, respectively, radius of the inscribed disc and the circumscribed disc Then for any disc of radius r r i r r e ), we have L 4πA + A L 4π A tan ) r 31) L 4πA + A L A tan r L 4πA + A π A ] ) tan r A tan r From the second formula of 31) we have 3) L 4πA + A A 1 4 tan r i and 33) L 4πA + A that is 34) L 1 tan r, e tan r e ) tan ri tan r e + tan r, i Theorem 6 Let D be a domain of area A and bonded by a simple closed curve of length L in the projective plane P R Let r i and r e r e π ) be, respectively, radius of the inscribed disc and the circumscribed disc Then we have L 4πA + A A 4 1 tan re 1 tan r i ) L 4πA + A L tan re tan ri tan r e +tan r i Each equality holds when and only when D is a geodesic disc

10 THE BONNESEN-TYPE INEQUALITIES IN A PLANE 1371 The Bonnesen-type inequalities in higher dimensional space are still unknown for many cases Zhang 8] has some results for convex domain D The Willmore functional inequalities of Bonnesen-type is investigated by the first author see 10, 11, 1, 13, 14, 15] for more details) Acknowledgement The work is partially supported by Hong Kong Qiu- Shi Science and Technologies Foundation and Southwest University of China Finally we wish to thank Professor Weiping Zhang, the Director of S S Chern Institute of Mathematics, for inviting us visiting the Institute several times in the past years We also like to thank referees for valuable comments and suggestions References 1] Y D Burago and V A Zalgaller, Geometric inequalities, Translated from the Russian by A B Sosinskiĭ Grundlehren der Mathematischen Wissenschaften, 85 Springer Series in Soviet Mathematics Springer-Verlag, Berlin, 1988 ] E Grinberg, D Ren, and J Zhou, The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature, preprint 3] E Grinberg, G Zhang, J Zhou, and S Li, Integral geometry and convexity, Proceedings of the 1st International Conference on Integral Geometry and Convexity Related Topics held at Wuhan University of Science and Technology, Wuhan, October 18 3, 004 Edited by Eric L Grinberg, Shougui Li, Gaoyong Zhang and Jiazu Zhou World Scientific Publishing Co Pte Ltd, Hackensack, NJ, 006 4] D Ren, Topics in integral geometry, Translated from the Chinese and revised by the author With forewords by Shiing Shen Chern and Chuan-Chih Hsiung Series in Pure Mathematics, 19 World Scientific Publishing Co, Inc, River Edge, NJ, ] L A Santaló, Integral geometry and geometric probability, With a foreword by Mark Kac Encyclopedia of Mathematics and its Applications, Vol 1 Addison-Wesley Publishing Co, Reading, Mass-London-Amsterdam, ] R Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, 44 Cambridge University Press, Cambridge, ] G Zhang, A sufficient condition for one convex body containing another, Chinese Ann Math Ser B ), no 4, ] G Zhang and J Zhou, Containment measures in integral geometry, Integral geometry and convexity, , World Sci Publ, Hackensack, NJ, 006 9] J Zhou, On Bonnesen-type inequalities, Acta Math Sin 50, No 6, ], When can one domain enclose another in R 3?, J Austral Math Soc Ser A ), no, ], The sufficient condition for a convex body to enclose another in R 4, Proc Amer Math Soc ), no 3, ], Sufficient conditions for one domain to contain another in a space of constant curvature, Proc Amer Math Soc ), no 9, ], Total square mean curvature of hypersurfaces, preprint submitted 14], The Willmore functional and the containment problem in R 4, Sci China Ser A: Math ), no 3, ], On Willmore functional for submanifolds, Canad Math Bull ), no 3,

11 137 JIAZU ZHOU AND FANGWEI CHEN Jiazu Zhou School of Mathematics and Statistics Southwest University Chongqing , P R China address: zhoujz@swueducn jzhou0788@yahoocom Fangwei Chen School of Mathematics and Statistics Wuhan University Wuhan, Hubei 43007, P R China address: cfw-yy@16com