Skip to main content

On containment measure and the mixed isoperimetric inequality

Abstract

We first investigate whether for given convex domains K 0 , K 1 in the Euclidean plane, for any rotation α, there is a translation x so that x+α K 1 K 0 or x+α K 1 K 0 . Then, we estimate the mixed isoperimetric deficit Δ 2 ( K 0 , K 1 ) of domains K 0 and K 1 via the known kinematic formulas of Poincaré and Blaschke in integral geometry. We obtain the sufficient condition for domain K 0 to contain, or to be contained in, convex domain x+α K 1 . Finally, we obtain the mixed isoperimetric inequality and some Bonnesen-style mixed inequalities. Those Bonnesen-style mixed inequalities obtained are the known Bonnesen-style inequalities if one of the domains is a disc. As a direct consequence, we obtain the strengthened Bonnesen isoperimetric inequality.

MSC:52A10, 52A22.

1 Introductions and preliminaries

A set of points K in the Euclidean space R n is convex if for all x,yK and 0λ1, λx+(1λ)yK. The convex hull K of K is the intersection of all convex sets that contain K. The Minkowski sum of convex sets K and L is defined by

K+L={x+y:xK,yL},

and the scalar product of convex set K for λ0 is defined by

λK={λx:xK}.

A homothety of a convex set K is of the form x+λK for x R n , λ>0. A convex body is a compact convex set with nonempty interior. A domain is a set with nonempty interior.

One may be interested in the following strong containment problem: Whether for given convex domains K 0 and K 1 , there exists a translation x so that x+α K 0 K 1 or x+α K 1 K 0 for any rotation α. It should be noted that this containment problem is much stronger than Hadwiger’s one. Therefore, the strong containment problem could lead to general and fundamental geometric inequalities (cf. [19]).

The well-known classical isoperimetric problem says that the disc encloses the maximum area among all domains of fixed perimeters in the Euclidean plane R 2 .

Proposition 1 Let Γ be a simple closed curve of length P in the Euclidean plane R 2 , then the area A of the domain K enclosed by Γ satisfies

P 2 4πA0.
(1)

The equality sign holds if and only if Γ is a circle.

Its analytic proofs root back to centuries ago. One 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 (cf. [1, 35, 1053]).

The isoperimetric deficit

Δ 2 (K)= P 2 4πA
(2)

measures the difference between domain K of area A and perimeter P, and a disc of radius P/2π.

During the 1920s, Bonnesen proved a series of inequalities of the form

Δ 2 (K)= P 2 4πA B K ,
(3)

where the quantity B K is an invariant of geometric significance having the following basic properties:

  1. 1.

    B K is nonnegative;

  2. 2.

    B K is vanish only when K is a disc.

Many B K s are found during the past. The main interest is still focusing on those unknown invariants of geometric significance. See references [35, 12, 17, 23, 31, 32, 36] for more details. The following Bonnesen’s isoperimetric inequality is well known.

Proposition 2 Let K be a domain of area A, bounded by a simple closed curve of perimeter P in the Euclidean plane R 2 . Let r and R be, respectively, the maximum inscribed radius and minimum circumscribed radius of K. Then we have the following Bonnesen’s isoperimetric inequality:

P 2 4πA π 2 ( R r ) 2 ,
(4)

where the equality holds if and only if K is a disc.

Since for any domain K in R 2 , its convex hull K increases the area A and decreases the perimeter P , that is, A A and P P, then we have P 2 4πA P 2 4π A , that is, Δ 2 (K) Δ 2 ( K ). Therefore, the isoperimetric inequality and the Bonnesen-style inequality are valid for all domains in R 2 if these inequalities are valid for convex domains.

In this paper, we first investigate the stronger containment problem: Whether for given convex bodies K 0 , K 1 in the Euclidean plane R 2 , there is a translation x so that x+α K 0 K 1 or x+α K 1 K 0 for any rotation α. Then we investigate the mixed isoperimetric deficit Δ 2 ( K 0 , K 1 ) of domains K 0 and K 1 .

Definition 1 Let K 0 and K 1 be two domains of areas A 0 and A 1 , and of perimeters P 0 and  P 1 , respectively. Then the mixed isoperimetric deficit of K 0 and K 1 is defined as

Δ 2 ( K 0 , K 1 )= P 0 2 P 1 2 16 π 2 A 0 A 1 .
(5)

Since the convex hull K of a set K in the Euclidean plane R 2 decreases the circum perimeter and increases the area, we have

Δ 2 ( K 0 , K 1 )= P 0 2 P 1 2 16 π 2 A 0 A 1 P 0 2 P 1 2 16 π 2 A 0 A 1 = Δ 2 ( K 0 , K 1 ) .

Therefore, we can only consider the convex domains when we estimate the mixed isoperimetric deficit low bound.

Via the kinematic formulas of Poincaré and Blaschke in integral geometry, we obtain sufficient conditions for convex domain K 1 to contain, or to be contained in, another convex domain K 0 for a translation x and any rotation α (Theorem 1 and Theorem 2). We obtain the mixed isoperimetric inequality and some Bonnesen-style mixed inequalities (Theorem 3, Theorem 4, Corollary 2, Corollary 3, Corollary 4, Theorem 5 and Theorem 6). One immediate consequence of our results is the strengthening Bonnesen isoperimetric inequality (Corollary 3). These new Bonnesen-style mixed inequalities obtained are fundamental and generalize some known Bonnesen-style inequalities (Corollary 5).

2 The containment measure

Let K k (k=0,1) be two domains of areas A k with simple boundaries of perimeters P k in the Euclidean plane R 2 . Let dg denote the kinematic density of the group G 2 of rigid motions, that is, translations and rotations, in R 2 . Let K 1 be convex, and let t K 1 (t(0,+)) be a homothetic copy of K 1 , then we have the known kinematic formula of Poincaré (cf. [3, 36])

{ g G 2 : K 0 t ( g K 1 ) } n { K 0 t ( g K 1 ) } dg=4t P 0 P 1 ,
(6)

where n{ K 0 t(g K 1 )} denotes the number of points of intersection K 0 t(g K 1 ).

Let m n be the kinematic measure of the set of positions g, for which t(g K 1 ) has exactly n intersection points with K 0 , i.e., m n =m{g G 2 :n{( K 0 )t(g K 1 )}=n}. Notice that the measure m n =0 for the odd n, then the formula of Poincaré can be rewritten as

n = 1 (2n) m 2 n =4t P 0 P 1 ,

that is,

n = 1 n m 2 n =2t P 0 P 1 .
(7)

We consider the homothetic copy t K 1 (t(0,+)) of K 1 .

Let χ( K 0 t(g K 1 )) be the Euler-Poincaré characteristics of the intersection K 0 t(g K 1 ). From the Blaschke’s kinematic formula (cf. [3, 36]):

{ g G 2 : K 0 t ( g K 1 ) } χ ( K 0 t ( g K 1 ) ) dg=2π ( t 2 A 1 + A 0 ) +t P 0 P 1 ,
(8)

we have

n = 1 m 2 n =2π ( t 2 A 1 + A 0 ) +t P 0 P 1 .
(9)

The formula of Poincaré (7) and the formula of Blaschke (9) give

n = 2 m 2 n (n1)=t P 0 P 1 2π ( t 2 A 1 + A 0 ) .

Since all m k are non-negative, we have

t P 0 P 1 2π ( t 2 A 1 + A 0 ) 0;t(0,+).
(10)

On the other hand, since domains K k (k=0,1) are assumed to be simply connected and bounded by simple curves, we have χ( K 0 t(g K 1 ))=n(g)= the number of connected components of the intersection K 0 g(t K 1 ). The fundamental kinematic formula of Blaschke (8) can be rewritten as

{ g G 2 : K 0 t ( g K 1 ) } n(g)dg=2π ( t 2 A 1 + A 0 ) +t P 0 P 1 .
(11)

If μ denotes set of all positions of K 1 , in which either t(g K 1 ) K 0 or t(g K 1 ) K 0 , then the above formula of Blaschke can be rewritten as

μ dg+ { g G 2 : K 0 t ( g K 1 ) } n(g)dg=2π ( t 2 A 1 + A 0 ) +t P 0 P 1 .
(12)

When K 0 t(g K 1 ), each component of K 0 t(g K 1 ) is bounded by at least an arc of K 0 and an arc of t(g K 1 ). Therefore, n(g)n{ K 0 t(g K 1 )}/2. Then by formulas of Poincaré and Blaschke, we obtain

μ dg2π ( t 2 A 1 + A 0 ) t P 0 P 1 .
(13)

Therefore, this inequality immediately gives the following answer for the strong containment problem (cf. [19, 17, 36, 50, 5460]).

Theorem 1 Let K k (k=0,1) be two domains of areas A k with simple boundaries of perimeters P k in R 2 . Let K 1 be convex. A sufficient condition for t K 1 to contain, or to be contained in, another domain K 0 for a translation and any rotation, is

2π A 1 t 2 P 0 P 1 t+2π A 0 >0.
(14)

Moreover, if t 2 A 1 A 0 , then t K 1 contains K 0 .

As a direct consequence of Theorem 1, we have the following analog of Ren’s theorem (cf. [36, 50, 5860]).

Theorem 2 Let K k (k=0,1) be two convex domains with areas A k and perimeters P k . Denote by Δ 2 ( K k )= P k 2 4π A k the isoperimetric deficit of K k . Then a sufficient condition for t K 1 , a homothetic copy of the convex domain K 1 , to contain domain K 0 for a translation and any rotation, is

t P 1 P 0 > t 2 Δ 2 ( K 1 ) + Δ 2 ( K 0 ) .
(15)

Proof Condition (15) means that t P 1 > P 0 and

2π A 1 t 2 P 0 P 1 t+2π A 0 >0.
(16)

By Theorem 1, we conclude that t K 1 either contains K 0 or is contained in K 0 . This inequality also leads to

2π ( t 2 A 1 A 0 ) >t P 0 P 1 4π A 0 > P 0 2 4π A 0 = Δ 2 ( K 0 ).

The isoperimetric inequality guarantees that t 2 A 1 > A 0 . We complete the proof of the theorem. □

3 Bonnesen-style mixed inequalities

Let r 01 =max{t:t(g K 1 ) K 0 ,g G 2 }, the maximum inscribed radius of K 0 with respect to  K 1 , and R 01 =min{t:t(g K 1 ) K 0 ,g G 2 }, the minimum circum scribed radius of K 0 with respect to K 1 . Note that r 01 , R 01 are, respectively, the maximum inscribed radius, the minimum circum radius of K 0 when K 1 is the unit disc. It is obvious that r 01 R 01 . Therefore, for t[ r 01 , R 01 ] neither t K 1 contains K 0 nor it is contained in K 0 . Then by Theorem 1, we have the following.

Theorem 3 Let K k (k=0,1) be two convex domains with areas A k and perimeters P k . Then

2π A 1 t 2 P 0 P 1 t+2π A 0 0; r 01 t R 01 .
(17)

When K 1 is the unit disc, this reduces to the following known Bonnesen inequality (cf. [3, 9, 31, 36, 61]).

Corollary 1 Let K be a convex domain with a simple boundary ∂K of length P and area A. Denote by R and r, respectively, the radius of the minimum circumscribed disc and radius of the maximum inscribed disc of K. Then

π t 2 Pt+A0;rtR.
(18)

By the two special cases of inequality (17):

2π A 1 r 01 2 P 0 P 1 r 01 +2π A 0 0;2π A 1 R 01 2 P 0 P 1 R 01 +2π A 0 0,

we obtain the following.

Theorem 4 Let K k (k=0,1) be two convex domains in the Euclidean plane R 2 with areas A k and perimeters P k . If K 1 is convex, then

P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 A 1 2 ( R 01 r 01 ) 2 + [ 2 π A 1 ( R 01 + r 01 ) P 0 P 1 ] 2 ,

where the equality holds if and only if r 01 = R 01 , that is, K 0 and K 1 are discs.

Proof By inequalities (19), we have

8 π 2 A 0 A 1 8 π 2 A 1 2 r 01 2 4 π A 1 r 01 P 0 P 1 , 8 π 2 A 0 A 1 8 π 2 A 1 2 R 01 2 4 π A 1 R 01 P 0 P 1 , P 0 2 P 1 2 16 π 2 A 0 A 1 P 0 2 P 1 2 + 8 π 2 A 1 2 r 01 2 + 8 π 2 A 1 2 R 01 2 4 π A 1 r 01 P 0 P 1 4 π A 1 R 01 P 0 P 1 .

Since

P 0 2 P 1 2 + 8 π 2 A 1 2 r 01 2 + 8 π 2 A 1 2 R 01 2 4 π A 1 r 01 P 0 P 1 4 π A 1 R 01 P 0 P 1 = 4 π 2 A 1 2 r 01 2 + 4 π 2 A 1 2 R 01 2 8 π 2 A 1 2 r 01 R 01 + P 0 2 P 1 2 + 4 π 2 A 1 2 r 01 2 + 4 π 2 A 1 2 R 01 2 + 8 π 2 A 1 2 r 01 R 01 4 π A 1 r 01 P 0 P 1 4 π A 1 R 01 P i P j = 4 π 2 A 1 2 ( R 01 r 01 ) 2 + ( 2 π A 1 r 01 + 2 π A 1 R 01 P 0 P 1 ) 2 ,

therefore,

P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 A 1 2 ( R 01 r 01 ) 2 + [ 2 π A 1 ( r 01 + R 01 ) P 0 P 1 ] 2 .

We complete the proof of Theorem 4. □

The following Kotlyar’s inequality (cf. [3, 24]) is an immediate consequence of Theorem 4.

Corollary 2 (Kotlyar)

Let K k (k=0,1) be two domains in R 2 with areas A k and perimeters  P k . If K 1 is convex, then

P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 A 1 2 ( R 01 r 01 ) 2 ,
(19)

where the equality holds if and only if both K 0 and K 1 are discs.

Let K 1 be the unit disc, then Theorem 4 immediately leads to the following inequality that strengthens the Bonnesen isoperimetric inequality (4).

Corollary 3 Let K be a domain of area A, bounded by a simple closed curve of length P in the Euclidean plane R 2 . Let r and R be, respectively, the inscribed radius and circumscribed radius of K, then

P 2 4πA π 2 ( R r ) 2 + [ π ( R + r ) P ] 2 ,
(20)

where the equality holds if and only if K is a disc.

One immediate consequence of Theorem 4 is the following mixed isoperimetric inequality:

P 0 2 P 1 2 16 π 2 A 0 A 1 0,

where the equality holds if and only if K 0 and K 1 are discs.

One may wish to consider the following Bonnesen-style mixed inequality:

P 0 2 P 1 2 16 π 2 A 0 A 1 B K 0 , K 1 ,

where B K 0 , K 1 is an invariant of K 0 and K 1 . B K 0 , K 1 is, of course, assumed to be nonnegative and vanishes only when both K 0 and K 1 are discs.

The inequality (17) can be rewritten as the following several inequalities:

P 0 2 P 1 2 16 π 2 A 0 A 1 ( P 0 P 1 4 π A 1 t ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( P 0 P 1 4 π A 0 t ) 2 ; r 01 t R 01 , P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 ( A 0 t A 1 t ) 2 ,
(21)

Therefore, we obtain the following Bonnesen-style mixed inequalities.

Corollary 4 Let K k (k=0,1) be two convex domains in the Euclidean plane R 2 with areas A k and perimeters P k . Then for r 01 t R 01 , we have

P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 A 1 2 ( R 01 t ) 2 + [ 2 π A 1 ( t + R 01 ) P 0 P 1 ] 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 A 1 2 ( t r 01 ) 2 + [ 2 π A 1 ( r 01 + t ) P 0 P 1 ] 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( P 0 P 1 4 π A 1 r 01 ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( 4 π A 0 r 01 P 0 P 1 ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 ( A 0 r 01 A 1 r 01 ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( P 0 P 1 4 π A 1 t ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( P 0 P 1 4 π A 0 t ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 ( A 0 t A 1 t ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( 4 π A 1 R 01 P 0 P 1 ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 ( P 0 P 1 4 π A 0 R 01 ) 2 ; P 0 2 P 1 2 16 π 2 A 0 A 1 4 π 2 ( A 1 R 01 A 0 R 01 ) 2 .
(22)

Each inequality holds as an equality if and only if both K 0 and K 1 are discs.

On the other hand, let us consider the following Bonnesen quadratic polynomial

B K 0 , K 1 (t)=2π A 1 t 2 P 0 P 1 t+2π A 0 .

It is clear that B K 0 , K 1 (0)>0 and B K 0 , K 1 (+)>0. If K 1 is convex, then the mixed isoperimetric inequality guarantees that two roots P 0 P 1 ± Δ 2 ( K 0 , K 1 ) 4 π A 1 of B K 0 , K 1 (t)=0 exist and satisfy

0< P 0 P 1 Δ 2 ( K 0 , K 1 ) 4 π A 1 r 01 R 01 P 0 P 1 + Δ 2 ( K 0 , K 1 ) 4 π A 1 <+.
(23)

The condition for existence of root(s) of the Bonnesen quadratic equation B K 0 , K 1 (t)=0 is the following symmetric mixed isoperimetric inequality:

Δ 2 ( K 0 , K 1 )= P 0 2 P 1 2 16 π 2 A 0 A 1 0.
(24)

The Bonnesen function B K 0 , K 1 (t)=2π A 1 t 2 P 0 P 1 t+2π A 0 attains minimum value Δ 2 ( K 0 , K 1 ) 8 π A 1 at t= P 0 P 1 4 π A 1 . The Bonnesen quadratic trinomial has only one root when Δ 2 ( K 0 , K 1 )=0. This means that both K 0 and K 1 are discs. This immediately leads to the following results.

Theorem 5 Let K k (k=0,1) be two convex domains of areas A k and perimeters P k in R 2 . Then

2π A 1 t 2 P 0 P 1 t+2π A 0 Δ 2 ( K 0 , K 1 ) 8 π A 1 .
(25)

Theorem 6 Let K k (k=0,1) be two convex domains of areas A k and perimeters P k in the Euclidean plane R 2 . Then we have

P 0 P 1 Δ 2 ( K 0 , K 1 ) 4 π A 1 r 01 P 0 P 1 4 π A 1 R 01 P 0 P 1 + Δ 2 ( K 0 , K 1 ) 4 π A 1 .
(26)

Each equality holds if and only if K 0 and K 1 are discs.

The following known Bonnesen-style inequalities are immediate consequences of Corollary 4, Theorem 5 and Theorem 6 when letting K 1 be the unit disc (cf. [3, 9, 12, 23, 31, 32, 36, 58, 62]).

Corollary 5 Let K be a plane domain of area A, bounded by a simple closed curve of length P. Let r and R be, respectively, the in-radius and out-radius of K. Then for any disc of radius t, rtR, we have the following Bonnesen-style inequalities:

P 2 4 π A ( P 2 π t ) 2 ; P 2 4 π A π 2 ( t r ) 2 + [ π ( t + r ) P ] 2 ; P 2 4 π A π 2 ( R t ) 2 + [ π ( R + t ) P ] 2 ; P 2 4 π A ( P 2 A t ) 2 ; P 2 4 π A ( A t π t ) 2 ; P 2 4 π A A 2 ( 1 r 1 R ) 2 ; P 2 4 π A P 2 ( R r R + r ) 2 ; P 2 4 π A A 2 ( 1 r 1 t ) 2 ; P 2 4 π A P 2 ( t r t + r ) 2 ; P 2 4 π A A 2 ( 1 t 1 R ) 2 ; P 2 4 π A P 2 ( R t R + t ) 2 ; P P 2 4 π A 2 π r t R P + P 2 4 π A 2 π .
(27)

Each equality holds if and only if K is a disc.

It should be noted that the first inequality in (27) is due to Bonnesen, and he only derived some inequalities for 2-dimensional case and never had any progress for higher dimensions or 2-dimensional surface of constant curvature. One would be interested in the situations in higher dimensional space R n and in the surface of constant curvature. Related development in those areas can be found in [26, 35, 37, 6366] and [58]. More details for the isoperimetric inequality and Bonnesen style inequalities can be found in [6780].

References

  1. Hadwiger H: Die isoperimetrische Ungleichung in Raum. Elem. Math. 1948, 3: 25–38.

    MathSciNet  MATH  Google Scholar 

  2. Hadwiger H: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin; 1957.

    Book  MATH  Google Scholar 

  3. Santaló LA: Integral Geometry and Geometric Probability. Addison-Wesley, Reading; 1976.

    MATH  Google Scholar 

  4. Zhou J: A kinematic formula and analogous of Hadwiger’s theorem in space. Contemp. Math. 1992, 140: 159–167.

    Article  MATH  Google Scholar 

  5. Zhou J:The sufficient condition for a convex body to enclose another in R 4 . Proc. Am. Math. Soc. 1994, 121(3):907–913.

    MathSciNet  MATH  Google Scholar 

  6. Zhou J:Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger’s theorem in R 2 n . Trans. Am. Math. Soc. 1994, 345(1):243–262.

    MATH  Google Scholar 

  7. Zhou J:When can one domain enclose another in R 3 . J. Aust. Math. Soc. A 1995, 59(2):266–272. 10.1017/S1446788700038660

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhou J: Sufficient conditions for one domain to contain another in a space of constant curvature. Proc. Am. Math. Soc. 1998, 126: 2797–2803. 10.1090/S0002-9939-98-04369-X

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhou J: On Bonnesen-type inequalities. Acta Math. Sinica (Chin. Ser.) 2007, 50(6):1397–1402.

    MathSciNet  MATH  Google Scholar 

  10. Banchoff TF, Pohl WF: A generalization of the isoperimetric inequality. J. Differ. Geom. 1971, 6: 175–213.

    MathSciNet  MATH  Google Scholar 

  11. Bokowski J, Heil E: Integral representation of quermassintegrals and Bonnesen-style inequalities. Arch. Math. 1986, 47: 79–89. 10.1007/BF01202503

    MathSciNet  Article  MATH  Google Scholar 

  12. Burago YD, Zalgaller VA: Geometric Inequalities. Springer, Berlin; 1988.

    Book  MATH  Google Scholar 

  13. Chen W, Howard R, Lutwak E, Yang D, Zhang G: A generalized affine isoperimetric inequality. J. Geom. Anal. 2004, 14(4):597–612. 10.1007/BF02922171

    MathSciNet  Article  MATH  Google Scholar 

  14. Croke C: A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv. 1984, 59(2):187–192.

    MathSciNet  Article  MATH  Google Scholar 

  15. Diskant V: A generalization of Bonnesen’s inequalities. Sov. Math. Dokl. 1973, 14: 1728–1731. (Transl. of Dokl. Akad. Nauk SSSR 213 (1973))

    MATH  Google Scholar 

  16. Enomoto K:A generalization of the isoperimetric inequality on S 2 and flat tori in S 3 . Proc. Am. Math. Soc. 1994, 120(2):553–558.

    MathSciNet  MATH  Google Scholar 

  17. Grinberg, E, Ren, D, Zhou, J: The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature. Preprint

  18. Grinberg E: Isoperimetric inequalities and identities for k -dimensional cross-sections of convex bodies. Math. Ann. 1991, 291: 75–86. 10.1007/BF01445191

    MathSciNet  Article  MATH  Google Scholar 

  19. Grinberg E, Zhang G: Convolutions, transforms, and convex bodies. Proc. Lond. Math. Soc. 1999, 78: 77–115. 10.1112/S0024611599001653

    MathSciNet  Article  MATH  Google Scholar 

  20. Gysin L: The isoperimetric inequality for nonsimple closed curves. Proc. Am. Math. Soc. 1993, 118(1):197–203. 10.1090/S0002-9939-1993-1079698-X

    MathSciNet  Article  MATH  Google Scholar 

  21. Howard R: The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces. Proc. Am. Math. Soc. 1998, 126: 2779–2787. 10.1090/S0002-9939-98-04336-6

    Article  MathSciNet  MATH  Google Scholar 

  22. Hsiang WY: An elementary proof of the isoperimetric problem. Chin. Ann. Math., Ser. A 2002, 23(1):7–12.

    MathSciNet  MATH  Google Scholar 

  23. Hsiung CC: Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary. Ann. Math. 1961, 73(2):213–220.

    MathSciNet  Article  MATH  Google Scholar 

  24. Kotlyar BD: On a geometric inequality. Ukr. Geom. Sb. 1987, 30: 49–52.

    MathSciNet  MATH  Google Scholar 

  25. Ku H, Ku M, Zhang X: Isoperimetric inequalities on surfaces of constant curvature. Can. J. Math. 1997, 49: 1162–1187. 10.4153/CJM-1997-057-x

    MathSciNet  Article  MATH  Google Scholar 

  26. Li M, Zhou J: An upper limit for the isoperimetric deficit of convex set in a plane of constant curvature. Sci. China Math. 2010, 53(8):1941–1946. 10.1007/s11425-010-4018-3

    MathSciNet  Article  MATH  Google Scholar 

  27. Li P, Yau ST: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 1982, 69: 269–291. 10.1007/BF01399507

    MathSciNet  Article  MATH  Google Scholar 

  28. Lutwak E, Yang D, Zhang G:Sharp affine L p Sobolev inequality. J. Differ. Geom. 2002, 62: 17–38.

    MathSciNet  MATH  Google Scholar 

  29. Lutwak E, Yang D, Zhang G: A new ellipsoid associated with convex bodies. Duke Math. J. 2000, 104(3):375–390. 10.1215/S0012-7094-00-10432-2

    MathSciNet  Article  MATH  Google Scholar 

  30. Lutwak E: On the Blaschke-Santaló inequality. Ann. New York Acad. Sci. 440. Discrete Geometry and Convexity 1985, 106–112.

    Google Scholar 

  31. Osserman R: The isoperimetric inequality. Bull. Am. Math. Soc. 1978, 84: 1182–1238. 10.1090/S0002-9904-1978-14553-4

    MathSciNet  Article  MATH  Google Scholar 

  32. Osserman R: Bonnesen-style isoperimetric inequality. Am. Math. Mon. 1979, 86: 1–29. 10.2307/2320297

    MathSciNet  Article  MATH  Google Scholar 

  33. Pleijel A: On konvexa kurvor. Nord. Mat. Tidskr. 1955, 3: 57–64.

    MathSciNet  Google Scholar 

  34. Polya G, Szego G Annals of Mathematics Studies 27. In Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton; 1951.

    Google Scholar 

  35. Rajala K, Zhong X:Bonnesen’s inequality for John domains in R n . J. Funct. Anal. 2012, 263(11):3617–3640. 10.1016/j.jfa.2012.09.004

    MathSciNet  Article  MATH  Google Scholar 

  36. Ren D: Topics in Integral Geometry. World Scientific, Singapore; 1994.

    MATH  Google Scholar 

  37. Sangwine-Yager JR: Mixe volumes. A. In Handbook of Convex Geometry. Edited by: Gruber P, Wills J. North-Holland, Amsterdam; 1993:43–71.

    Chapter  Google Scholar 

  38. Sangwine-Yager JR: A Bonnesen-style in radius inequality in 3-space. Pac. J. Math. 1988, 134(1):173–178. 10.2140/pjm.1988.134.173

    MathSciNet  Article  MATH  Google Scholar 

  39. Schneider R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge; 1993.

    Book  MATH  Google Scholar 

  40. Stone A: On the isoperimetric inequality on a minimal surface. Calc. Var. Partial Differ. Equ. 2003, 17(4):369–391. 10.1007/s00526-002-0174-9

    Article  MathSciNet  MATH  Google Scholar 

  41. Tang D: Discrete Wirtinger and isoperimetric type inequalities. Bull. Aust. Math. Soc. 1991, 43: 467–474. 10.1017/S0004972700029312

    Article  MathSciNet  MATH  Google Scholar 

  42. Teissier B: Bonnesen-type inequalities in algebraic geometry, I. Introduction to the problem. In Seminar on Differential Geometry Edited by: Yau ST. 1982, 85–105.

    Google Scholar 

  43. Teufel E: A generalization of the isoperimetric inequality in the hyperbolic plane. Arch. Math. 1991, 57(5):508–513. 10.1007/BF01246751

    MathSciNet  Article  MATH  Google Scholar 

  44. Teufel E: Isoperimetric inequalities for closed curves in spaces of constant curvature. Results Math. 1992, 22: 622–630. 10.1007/BF03323109

    MathSciNet  Article  MATH  Google Scholar 

  45. Wei S, Zhu M: Sharp isoperimetric inequalities and sphere theorems. Pac. J. Math. 2005, 220(1):183–195. 10.2140/pjm.2005.220.183

    MathSciNet  Article  MATH  Google Scholar 

  46. Weiner JL: A generalization of the isoperimetric inequality on the 2-sphere. Indiana Univ. Math. J. 1974, 24: 243–248. 10.1512/iumj.1975.24.24021

    MathSciNet  Article  MATH  Google Scholar 

  47. Weiner JL: Isoperimetric inequalities for immersed closed spherical curves. Proc. Am. Math. Soc. 1994, 120(2):501–506. 10.1090/S0002-9939-1994-1163337-4

    MathSciNet  Article  MATH  Google Scholar 

  48. Yau ST: Isoperimetric constants and the first eigenvalue of a compact manifold. Ann. Sci. Éc. Norm. Super. 1975, 8(4):487–507.

    MathSciNet  MATH  Google Scholar 

  49. Yau ST: Sobolev inequality for measure space. In Tsing Hua Lectures on Geometry and Analysis. International Press, Cambridge; 1997:299–313.

    Google Scholar 

  50. Zhang G, Zhou J: Containment measures in integral geometry. In Integral Geometry and Convexity. World Scientific, Singapore; 2006:153–168.

    Chapter  Google Scholar 

  51. Zhang X-M: Bonnesen-style inequalities and pseudo-perimeters for polygons. J. Geom. 1997, 60: 188–201. 10.1007/BF01252226

    MathSciNet  Article  MATH  Google Scholar 

  52. Zhang X-M: Schur-convex functions and isoperimetric inequalities. Proc. Am. Math. Soc. 1998, 126(2):461–470. 10.1090/S0002-9939-98-04151-3

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhao L, Ma L, Zhou J: A geometric application of the Wirtinger inequality. J. Math. (Wuhan) 2011, 31(5):887–890.

    MathSciNet  MATH  Google Scholar 

  54. Zhou J: On Willmore inequality for submanifolds. Can. Math. Bull. 2007, 50(3):474–480. 10.4153/CMB-2007-047-4

    Article  MATH  Google Scholar 

  55. Zhou J: On the Willmore deficit of convex surfaces. Lectures in Applied Mathematics of Amer. Math. Soc. 30. Tomography, Impedance Imaging, and Integral Geometry 1994, 279–287.

    Google Scholar 

  56. Zhou J, Chen F: The Bonnesen-type inequalities in a plane of constant curvature. J. Korean Math. Soc. 2007, 44(6):1–10.

    MathSciNet  Article  MATH  Google Scholar 

  57. Zhou J:The Willmore functional and the containment problem in R 4 . Sci. China Ser. A 2007, 50(3):325–333. 10.1007/s11425-007-0029-0

    MathSciNet  Article  MATH  Google Scholar 

  58. Zhou J, Ren D: Geometric inequalities from the viewpoint of integral geometry. Acta Math. Sci. Ser. A Chin. Ed. 2010, 30(5):1322–1339.

    MathSciNet  MATH  Google Scholar 

  59. Zeng C, Zhou J, Yue S: The symmetric mixed isoperimetric inequality of two planar convex domains. Acta Math. Sin. 2012, 55(2):355–362.

    MathSciNet  MATH  Google Scholar 

  60. Zhou J, Ma L, Xu W: On the isoperimetric deficit upper limit. Bull. Korean Math. Soc. 2013, 50(1):175–184. 10.4134/BKMS.2013.50.1.175

    MathSciNet  Article  MATH  Google Scholar 

  61. Bonnesen T: Les probléms des isopérimétres et des isépiphanes. Gauthier-Villars, Paris; 1929.

    MATH  Google Scholar 

  62. Grinberg E, Li S, Zhang G, Zhou J: Integral geometry and convexity. In Proceedings of the 1st International Conference on Integral Geometry and Convexity Related Topics. World Scientific, Singapore; 2006.

    Chapter  Google Scholar 

  63. Fuglede B:Stability in the isoperimetric problem for convex or nearly spherical domains in R n . Trans. Am. Math. Soc. 1989, 314: 619–638.

    MathSciNet  MATH  Google Scholar 

  64. Fusco N, Gelli MS, Pisante G: On a Bonnesen type inequality involving the spherical deviation. J. Math. Pures Appl. 2012, 98(6):616–632.

    MathSciNet  Article  MATH  Google Scholar 

  65. Fusco N, Maggi F, Pratelli A: The sharp quantitative isoperimetric inequality. Ann. Math. 2008, 168(3):941–980. 10.4007/annals.2008.168.941

    MathSciNet  Article  MATH  Google Scholar 

  66. Zeng C, Ma L, Zhou J: The Bonnesen isoperimetric inequality in a surface of constant curvature. Sci. China Math. 2012, 55(9):1913–1919. 10.1007/s11425-012-4405-z

    MathSciNet  Article  MATH  Google Scholar 

  67. Blaschke W: Vorlesungen über Intergralgeometrie. 3rd edition. Deutsch. Verlag Wiss., Berlin; 1955.

    Google Scholar 

  68. Blaschke W: Kreis und Kugel. de Gruyter, Berlin; 1956.

    Book  MATH  Google Scholar 

  69. Bonnesen T, Fenchel W: Theorie der konvexen Köeper. 2nd edition. Springer, Berlin; 1974.

    Book  MATH  Google Scholar 

  70. Bottema O: Eine obere Grenze für das isoperimetrische Defizit ebener Kurven. Nederl. Akad. Wetensch. Proc. 1933, A66: 442–446.

    MATH  Google Scholar 

  71. Flanders H: A proof of Minkowski’s inequality for convex curves. Am. Math. Mon. 1968, 75: 581–593. 10.2307/2313773

    MathSciNet  Article  MATH  Google Scholar 

  72. Hardy G, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1951.

    MATH  Google Scholar 

  73. Howard R: Blaschke’s rolling theorem for manifolds with boundary. Manuscr. Math. 1999, 99(4):471–483. 10.1007/s002290050186

    Article  MATH  Google Scholar 

  74. Koutrouieiotis D: On Blaschke’s rolling theorems. Arch. Math. 1972, 21(1):655–660.

    Article  MATH  Google Scholar 

  75. Zhang G: The affine Sobolev inequality. J. Differ. Geom. 1999, 53: 183–202.

    MathSciNet  MATH  Google Scholar 

  76. Zhang G: Geometric inequalities and inclusion measures of convex bodies. Mathematika 1994, 41: 95–116. 10.1112/S0025579300007208

    MathSciNet  Article  MATH  Google Scholar 

  77. Zhang, G: Convex Geometric Analysis. Preprint

  78. Zhou J, Xia Y, Zeng C: Some new Bonnesen-style inequalities. J. Korean Math. Soc. 2011, 48(2):421–430. 10.4134/JKMS.2011.48.2.421

    MathSciNet  Article  MATH  Google Scholar 

  79. Zhou J, Du Y, Cheng F: Some Bonnesen-style inequalities for higher dimensions. Acta Math. Sin. 2012, 28(12):2561–2568. 10.1007/s10114-012-9657-6

    MathSciNet  Article  MATH  Google Scholar 

  80. Zhou J, Zhou C, Ma F: Isoperimetric deficit upper limit of a planar convex set. Rend. Circ. Mat. Palermo Suppl. 2009, 81: 363–367.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Authors would like to thank two anonymous referees for many helpful comments and suggestions that directly lead to the improvement of the original manuscript. The corresponding author is supported in part by the NSFC (No. 11271302) and the Ph.D. Program of Higher Education Research Fund (No. 2012182110020).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jiazu Zhou.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

Authors’ original file for figure 1

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Xu, W., Zhou, J. & Zhu, B. On containment measure and the mixed isoperimetric inequality. J Inequal Appl 2013, 540 (2013). https://doi.org/10.1186/1029-242X-2013-540

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-540

Keywords

  • convex set
  • containment measure
  • mixed isoperimetric deficit
  • mixed isoperimetric inequality
  • Bonnesen-style mixed inequality