THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN THE CONSTANT CURVATURE -SPACES MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ Abstract In this paper we prove several new inequalities for centrally symmetric convex bodies in the -dimensional spaces of constant curvature, which have their analog in the plane Thus, when tends to 0, the classical planar inequalities will be obtained For instance, we get the relation between the perimeter and the diameter of a symmetric convex body Rosenthal-Szasz inequality) which, together with the wellknown spherical/hyperbolic isoperimetric inequality, allows to solve the isodiametric problem The analogs to other classical planar relations are also proved 1 Introduction If K is a planar convex body, ie, a compact convex set in R, with area AK) and diameter DK), the well-known isodiametric inequality states that 1) πdk) 4AK), with equality if and only if K is a circle The isodiametric inequality can be obtained as a consequence of the famous isoperimetric inequality, ) pk) 4πAK), and the classical Rosenthal-Szasz s theorem, namely, 3) pk) πdk); here pk) denotes the perimeter of K In ) equality holds if and only if K is a circle, whereas for 3) all constant width sets verify the equality For detailed information on these classical inequalities we refer to [3, 10] The isoperimetric problem has its analog on the sphere S and the hyperbolic plane H with curvature 0: Bernstein [] respectively, Schmidt [13]) proved that if K is the region bounded by a convex curve on S respectively, H ), then 4) pk) 4πAK) AK), 000 Mathematics Subject Classification Primary 5A40, 5A55; Secondary 5A10 Key words and phrases Spherical/hyperbolic isodiametric inequality, Rosenthal-Szasz Theorem, spherical/hyperbolic symmetric convex bodies, minimal geodesic) width, perimeter, area, diameter Supported by MINECO-FEDER project MTM01-34037 and by Programa de Ayudas a Grupos de Excelencia de la Región de Murcia, Fundación Séneca, 04540/GERM/06 1
MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ with equality only for the geodesic discs see also, eg, [1, p 34]) However, just a few classical inequalities of the Euclidean plane have been translated into the sphere and the hyperbolic space for instance, Jung s inequality, see [4], or Bonnesen s inequality, see [10]) For other problems in the hyperbolic plane/space we refer, for instance, to [1, 5, 6, 7, 8] The aim of this paper is to consider other classical inequalities of planar convex bodies, looking for their analog on the constant curvature spaces under symmetry assumptions For instance, in the case of the sphere we get the following spherical isodiametric inequality: Theorem 11 Let K be a centrally symmetric convex body in S Then 4π ) DK) 5) sin 4πAK) AK), and equality holds if and only if K is a geodesic disc As in the case of the classical planar) isodiametric inequality, 5) will be obtained as a direct consequence of the spherical isoperimetric inequality 4) and an analog to Rosenthal-Szasz theorem 3) for the sphere; namely, we prove the following result: Theorem 1 Let K be a centrally symmetric convex body in S Then 6) pk) π ) DK) sin and equality holds if and only if K is a geodesic disc At this point we observe that if K is not centrally symmetric, then inequality 6) is not true, as an octant of the sphere shows easily We also notice that in the two above results, when tends to 0 we obtain the classical isodiametric inequality 1) and the Rosenthal-Szasz result 3), respectively Inequalities 5) and 6) provide upper bounds for the area and the perimeter of a centrally symmetric spherical convex body, respectively We also wonder, analogously to the planar case, whether they can be bounded from below We give a positive answer to this question, involving the minimal width ωk) of the convex body see Section 4 for the definition) Theorem 13 Let K be a centrally symmetric convex body in S Then pk) π ) ωk) sin, 7) AK) π [ 1 cos ωk) )] Equality holds in both inequalities if and only if K is a geodesic disc Theorems 11 and 1 are proved in Section 3, as well as other new inequalities eg a spherical Bonnesen-type isodiametric inequality for centrally symmetric convex bodies, see Proposition 33) In Section 4 the results
THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN H AND S 3 involving the minimal width, namely, Theorem 13 and other relations, are obtained First, Section is devoted to introduce the notation, definitions and some preliminary results which will be needed along the proofs In the case of the hyperbolic plane H, since the proofs of the corresponding results are analogous to the ones in the sphere, for the sake of brevity we just state the inequalities for H in an appendix at the end of the paper Notation for the case of the sphere and preliminaries Let S be the -dimensional sphere of curvature > 0, ie, with radius 1/ A subset K S is said to be convex in S ) if it satisfies the following two properties: K is contained in an open hemisphere; for any p, q K, there exists a unique arc length parametrized) geodesic segment, ie, segment of great circle, joining p and q, which is contained in K We observe that uniqueness comes from the assumption that K is contained in an open hemisphere From now on we will represent by γ q p the geodesic starting at p and passing through q, and for the sake of brevity, we will write γ q p to denote the geodesic) segment of γ q p between p and q The set of all convex bodies, ie, compact convex sets with non-empty interior, in the sphere S will be denoted by KS ), and for K KS ), we denote by AK) and pk) its area Lebesgue measure) and perimeter length of the boundary curve), respectively We notice that for any K KS ), AK) > 0 Finally, bd K and int K will represent the boundary and the interior in S ) of K The intrinsic distance d : K K [ 0, DK) ] is given by dp, q) = Lγ q p), ie, the length of the geodesic segment γ q p As usual in the literature, for p S we represent the exponential map at p by exp p : T p S S, and for any v T p S, we write γ p,v t) to denote the geodesic passing through p with velocity vector v, ie, γ p,v t) = exp p tv), t 0 The geodesic disc centered at p and with radius r 0, namely, Dp, r) = exp p {v Tp S : v r} ), coincides with the closed ball B d p, r) in the intrinsic distance see [9, Proposition 76]) Its perimeter and area take the values see, eg, [14, p 85]) p Dp, r) ) = π sin r ) and 8) A Dp, r) ) = π [ )] 1 cos r Clearly, lim 0 p Dp, r) ) = πr and lim 0 A Dp, r) ) = πr, ie, the perimeter and the area of a planar disc of radius r For K KS ), the diameter DK) is defined as the maximum intrinsic distance between two points of K, its circumradius RK) is the greatest lower bound of all radii R such that K is contained in a geodesic disc of
4 MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ radius R, and the inradius rk) is the least upper bound of all radii r such that K contains a geodesic disc of radius r 1 On the monotonicity of the area and the perimeter On the one hand, it is a well-known fact that the Lebesgue measure in the sphere indeed any positive measure) is a monotonous functional, ie, if K, L KS ) with K L, then AK) AL) On the other hand, in [10, Proposition 13], the monotonicity of the perimeter in KS ) was shown: Lemma 1 [10, Proposition 13]) Let K, L KS ) such that K L Then pk) pl) The proof of this fact is based on the so-called Principal Kinematic Formula for S see eg [1, p 31]) Next we provide a characterization of the equality cases, which will be needed for the proofs of the equality cases in the forthcoming results Lemma Let K, L KS ), K L i) If AK) = AL) then K = L ii) If pk) = pl) then K = L Proof In order to prove i), we assume that K L Then, on the one hand, there exist p int L) L\K), and hence r > 0 small enough such that Dp, r) L\K Indeed, if p L\K and p int L, it would be p bd L; then since K, L are compact and convex, the spherical) convex hull ie, the smallest convex body in the sphere containing the set) conv {p} K ) L and conv {p} K ) \K would contain interior points of L Therefore, by the monotonicity of the area, AL\K) A Dp, r) ) > 0 On the other hand, since L = K L\K), the additivity of the area functional yields AL) = AK) + AL\K), and using the assumption AL) = AK) we get AL\K) = 0, a contradiction Next we show ii) Without loss of generality, and for the sake of brevity, we fix = 1, and we assume that pk) = pl) and K L Then there exists a point p bd L)\K Moreover, since K is compact, there exists q bd K such that dp, q) = min { dp, r) : r bd K } Let C = pos K be the positive hull in R 3 of K and let H be the unique) supporting plane in R 3 of C at q see Figure 1) Denoting by H + the half-space determined by H and containing C, and taking Q = L H +, it is clear that K Q L Moreover, since H is a plane through the origin, Q H = γ q q 1 is a geodesic segment with, say, end points q 1, q We observe, on the one hand, that since L is convex, then q 1, q cannot be antipodal points, and thus the curve segment α in bd L joining q 1 and q and passing through p) is not a geodesic segment see Figure 1)
THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN H AND S 5 Figure 1 The set Q and the curves joining q 1, q Therefore, Lγ q q 1 ) < Lα), because the intrinsic distance between two points in the same open hemisphere is minimized by geodesic segments, and since p / H +, we finally get 9) pq) < pl) On the other hand, by Lemma 1 we have that 10) pk) pq) Then, 9) and 10), together with our hypothesis, lead to which is a contradiction pk) pq) < pl) = pk), Spherical centrally symmetric convex bodies Next we consider symmetric convex bodies in the sphere A spherical convex body K KS ) is said to be centrally symmetric, if there exists a point c K K such that for all p bd K, dc K, p) = dc K, p ), where p = γ c K p bd K, p p see Figure ) Figure A centrally symmetric convex body K in S with center c K Next lemma shows that the point c K is unique, which we call the center of K in S The point p is the symmetric point of p with respect to c K Lemma 3 Let K KS ) be centrally symmetric Then the center c K is unique
6 MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ Proof Let c 1, c be two centers of K and for the sake of brevity we write γ = γ c c 1 Let {p, q} = γ bd K Without loss of generality we may assume that the points p, c 1, c, q are ordered on γ as shown in Figure 3, ie, if x = γt x ), for x {p, q, c 1, c }, then t p < t c1 < t c < t q Figure 3 The center c K of a centrally symmetric convex body K KS ) is unique On the one hand, since c 1, c are centers of K, we have that dp, c 1 ) = dc 1, q) = dp, c ) = dc, q) = dp, q) On the other hand, since c 1, c, q lie on the same minimizing) geodesic, then dc 1, q) = dc 1, c ) + dc, q), and thus we get dc 1, c ) = 0 Therefore c 1 = c We notice that if K = Dp, r) KS ) is a geodesic disc, then c K is just the usual center p of K 3 The spherical isodiametric inequality We start this section showing Theorem 1, which establishes a Rosenthal- Szasz-type inequality in S for centrally symmetric convex bodies Then, Theorem 11 will be a direct consequence of the spherical isoperimetric inequality 4) and 6) Proof of Theorem 1 First we show that 11) K B d c K, DK) ) Indeed, let z K and we assume that dz, c K ) > DK)/ Then, denoting by z the symmetral of z with respect to c K see Figure 4), we have that dz, z ) = dz, c K ) + dc K, z ) > DK) + DK) = DK), which contradicts the fact that DK) is the diameter of K Thus dz, c K ) DK)/, ie, z B d ck, DK)/ )
THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN H AND S 7 Figure 4 If K is symmetric then K B d ck, DK)/ ) Then, Lemma 1 together with 8) yields )) pk) p B d c K, DK) = π ) DK) sin Finally, if pk) = p B d ck, DK)/ )), since K B d ck, DK)/ ), then Lemma ensures that K = B d ck, DK)/ ) is a geodesic disc It is easy to check that when goes to 0 in 6), the classical inequality of Rosenthal and Szasz 3) is obtained We observe that if K KS ) is centrally symmetric, then the relation DK) = RK) holds Indeed, from the inclusion 11) and the definition of circumradius, we immediately get RK) DK) Moreover, for arbitrary K KS ), there exists p 0 S such that K B d p0, RK) ) Then, taking p, q K B d p0, RK) ), we obtain dp, q) dp, p 0 ) + dp 0, q) RK), and therefore, DK) RK) We also notice that inequality 6) fails if the symmetry assumption is removed, as an octant of S shows In that case, the following proposition provides a not sharp) bound for the perimeter in terms of the diameter Proposition 31 Let K KS ) Then pk) 4π ) DK) sin 3 Proof By definition of circumradius there holds K D p, RK) ) for some p K, and hence, using the monotonicity of the perimeter see Lemma 1) and 8), we get pk) p D p, RK) )) = π sin RK) ) Finally, we use the spherical Jung inequality DK) n + 1 arcsin n sin RK) ) ), for n = see [4, Theorem ]) in order to bound the perimeter in terms of the diameter
8 MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ In the case of arbitrary not necessarily centrally symmetric) convex bodies of the sphere, Proposition 31 and the spherical isoperimetric inequality 4) yield the following isodiametric relation for arbitrary convex bodies K KS ): Corollary 3 Let K KS ) Then 16π ) DK) 3 sin 4πAK) AK) We notice that inequalities in Proposition 31 and Corollary 3 are not sharp On the other hand, a series of inequalities of the form pk) 4πAK) F K) where proved by Bonnesen during the 190 s, where F K) is a geometric non-negative functional, which vanishes only if K is a circle Perhaps the most famous Bonnesen inequality is provided by the planar, classical) circumradius and inradius: 1) pk) 4πAK) π RK) rk) ) Thus, using 3), Bonnesen s inequality 1) easily provides also a bound for the isodiametric deficit πdk) 4AK), namely, 13) πdk) 4AK) π RK) rk) ), which we call a Bonnesen-type isodiametric inequality Here equality also holds if and only if K is a circle In [10, Theorem 5], the following spherical Bonnesen isoperimetric) inequality has been proved: for K KS ) it holds pk) 4πAK) AK) ) 14) 1 [ sin RK) ) sin rk) )] ) π AK) 4 Thus, using now Theorem 1, the spherical Bonnesen inequality 14) leads to a Bonnesen-type isodiametric inequality for centrally symmetric convex bodies in the sphere S : Proposition 33 Let K KS ) be centrally symmetric Then 4π ) DK) sin 4πAK) AK) ) 1 [ sin RK) ) sin rk) )] ), π AK) 4 with equality if and only if K is a geodesic disc We notice that π AK) > 0 for all K KS ) because K is contained in an open hemisphere Therefore, the spherical isodiametric inequality Theorem 11) can be also obtained as a direct consequence of the above proposition
THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN H AND S 9 4 Relating the perimeter and the area to the minimal width in the sphere The width of a spherical convex body can be defined in the following way see eg [6]) Let K KS ) and let p bd K be a regular point, ie, the unit) tangent vector vp) to bd K at p is unique Let np) T p S be the inner normal unit vector to the curve bd K at p, ie, let np) = p vp) p vp) = [ p vp) ], if it points to the interior of K, or [ vp) p ] on the contrary, and we take the geodesic γ p,np) see Figure 5) Figure 5 The width of a convex body K at p bd K Next we consider the family γ t of geodesics which are orthogonal to γ p,np) at each point γ p,np) t), t 0 Then, there exists the smallest positive number t 0 > 0 such that the geodesic γ t0 passing through the point γ p,np) t 0 ) is tangent to bd K, and clearly, γ p,vp) and γ t0 are two supporting geodesics to bd K, orthogonal to γ p,np), and such that K is contained in the strip determined by them The width ωk, p) of K at p bd K is then defined as the length of γ p,np) between the points p = γ p,np) 0) and γ p,np) t 0 ), ie, ωk, p) = d p, γ p,np) t 0 ) ) = t 0 If p bd K is not regular, we do the same construction for all tangent vectors vp) to bd K at p, obtaining in this way a width-value ω K, p, vp) ) for each vector vp); then we define the width ωk, p) of K at p bd K as the minimum min vp) ω K, p, vp) ) Finally the minimal width of K is defined as ωk) = min { ωk, p) : p bd K } We would like to point out the similarity of this notion with the classical definition of minimal width of a planar ie, = 0) convex body minimum distance between two supporting lines to K)
10 MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ Next we show the relation between the perimeter/area and the minimal width Proof of Theorem 13 First we show the inclusion 15) B d c K, ωk) ) K We reason by contradiction assuming that there exists a point z B d c K, ωk) ) \K Let z 0 = bd K γ c K z and let p bd K be a boundary point of K such that dc K, p) attains the minimum see Figure 6) Figure 6 If K is symmetric then K B d ck, ωk)/ ) Then see, eg, [11, p 80, Corollary 6]) the geodesic γ c K p and the curve bd K are orthogonal at p, and thus ωk, p) = dp, p ) = dc K, p) dc K, z 0 ) < dc K, z) ωk), a contradiction Therefore B d ck, ωk)/ ) K, and using the monotonicity of the area and the perimeter see Lemma 1), together with 8), we obtain that )) AK) A B d c K, ωk) = π [ )] ωk) 1 cos and )) pk) p B d c K, ωk) = π ) ωk) sin Finally, if the equality pk) = p B d ck, ωk)/ )) holds, or analogously, if AK) = A B d ck, ωk)/ )), since K B d ck, ωk)/ ), then Lemma ensures that K = B d ck, DK)/ ) is a geodesic disc We notice that when tends to 0, the classical relation pk) πωk)
THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN H AND S 11 of the plane is obtained see eg [3, p 84]) Moreover, the area inequality in 7) yields AK) π 4 ωk) if 0, a known lower bound for the ratio AK)/ωK) for the case of centrally symmetric sets see eg [3, p 83]) We also observe that the expected relation ωk) DK) holds: indeed, from 15) and the definition of inradius, we immediately get ωk) rk), and thus ωk) rk) RK) = DK) 5 Appendix: the hyperbolic case Let H be the -dimensional hyperbolic space of curvature < 0, for which we can consider, eg, the Beltrami model A subset K H is said to be convex in H ) if for any p, q K, there exists a unique arc length parametrized) geodesic segment joining p and q, which is contained in K The set of all convex bodies in H will be denoted by KH ) The other definitions are analogous to the ones in the sphere Now, the perimeter and the area of a geodesic disc in H take the values p Dp, r) ) = π sinh r ) and A Dp, r) ) = π [ 1 cosh r )] The monotonicity of the perimeter and the area Lemma 1, see [10, Proposition 13]) with the equality cases Lemma ), as well as the uniqueness of the center of symmetry Lemma 3) also hold Thus, analogous proofs to the ones of the results in the sphere allow to show the following results: Theorem 51 Hyperbolic isodiametric inequality) Let K KH ) be centrally symmetric Then ) 4π DK) sinh 4πAK) AK), and equality holds if and only if K is a geodesic disc Theorem 5 Hyperbolic Rosenthal-Szasz theorem) Let K KH ) be centrally symmetric Then pk) π ) DK) sinh and equality holds if and only if K is a geodesic disc
1 MARÍA A HERNÁNDEZ CIFRE AND ANTONIO R MARTÍNEZ FERNÁNDEZ Theorem 53 Let K KH ) be centrally symmetric Then pk) π ) ωk) sinh, AK) π [ ) ] ωk) cosh 1 Equality holds in both inequalities if and only if K is a geodesic disc Proposition 54 Let K KH ) Then pk) 4π ) DK) sinh 3 Corollary 55 Let K KH ) Then ) 16π DK) 3 sinh 4πAK) AK) Acknowledgement The authors would like to thank the anonymous referees for the very valuable comments and helpful suggestions References [1] J W Anderson, Hyperbolic geometry Second edition Springer Undergraduate Mathematics Series Springer-Verlag London, Ltd, London, 005 [] F Bernstein, Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math Ann 60 1) 1905), 117-136 [3] T Bonnesen and W Fenchel, Theorie der konvexen Körper Springer, Berlin, 1934, 1974 English translation: Theory of convex bodies Edited by L Boron, C Christenson and B Smith BCS Associates, Moscow, ID, 1987 [4] B V Dekster, The Jung theorem for spherical and hyperbolic spaces, Acta Math Hungar 67 4) 1995), 315 331 [5] E Gallego, A Reventós, Asymptotic behavior of convex sets in the hyperbolic plane, J Differential Geom 1 1985), 63-7 [6] E Gallego, A Reventós, G Solanes, E Teufel, Width of convex bodies in spaces of constant curvature, Manuscripta Math 16 008), 115 134 [7] E Gallego, G Solanes, Perimeter, diameter and area of convex sets in the hyperbolic plane, J London Math Soc 64 1) 001), 161-178 [8] E Gallego, G Solanes, Integral geometry and geometric inequalities in hyperbolic space, Differential Geom Appl 3) 005), 315-35 [9] M A Hernández Cifre, J A Pastor, Lectures on Differential Geometry Spanish) Publicaciones del CSIC, Textos Universitarios 47, Madrid, 010 [10] D A Klain, Bonnesen-type inequalities for surfaces of constant curvature, Adv Appl Math 39 ) 007), 143 154 [11] B O neill, Semi-Riemannian Geometry With Applications to Relativity Pure and Applied Mathematics, 103 Academic Press, Inc, New York, 1983 [1] L A Santaló, Integral Geometry and Geometric Probability Encyclopedia of Mathematics and its Applications, 1 Addison-Wesley Publishing Co, Reading, Mass- London-Amsterdam, 1976 [13] E Schmidt, Über die isoperimetrische Aufgabe im n-dimensionalen Raum konstanter negativer Krümmung I Die isoperimetrischen Ungleichungen in der hyperbolischen
THE ISODIAMETRIC PROBLEM AND OTHER INEQUALITIES IN H AND S 13 Ebene und für Rotationskörper im n-dimensionalen hyperbolischen Raum, Math Z 46 1940), 04-30 [14] J Stillwell, Geometry of Surfaces Springer-Verlag, New York, 199 Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain E-mail address: mhcifre@umes E-mail address: antoniorobertomartinez@umes