Open Access

On containment measure and the mixed isoperimetric inequality

Journal of Inequalities and Applications20132013:540

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

Received: 19 December 2012

Accepted: 30 August 2013

Published: 19 November 2013

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.

Keywords

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

1 Introductions and preliminaries

A set of points K in the Euclidean space R n is convex if for all x , y K and 0 λ 1 , λ x + ( 1 λ ) y K . 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 : x K , y L } ,
and the scalar product of convex set K for λ 0 is defined by
λ K = { λ x : x K } .

A homothety of a convex set K is of the form x + λ K 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 4 π A 0 .
(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 4 π A
(2)

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

During the 1920s, Bonnesen proved a series of inequalities of the form
Δ 2 ( K ) = P 2 4 π A 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 4 π A 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 4 π A P 2 4 π 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 ) } d g = 4 t 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 ( 2 n ) m 2 n = 4 t P 0 P 1 ,
that is,
n = 1 n m 2 n = 2 t 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 ) ) d g = 2 π ( t 2 A 1 + A 0 ) + t P 0 P 1 ,
(8)
we have
n = 1 m 2 n = 2 π ( 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 ( n 1 ) = t P 0 P 1 2 π ( t 2 A 1 + A 0 ) .
Since all m k are non-negative, we have
t P 0 P 1 2 π ( 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 ) d g = 2 π ( 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 d g + { g G 2 : K 0 t ( g K 1 ) } n ( g ) d g = 2 π ( 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 d g 2 π ( 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
2 π A 1 t 2 P 0 P 1 t + 2 π 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 4 π 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
2 π A 1 t 2 P 0 P 1 t + 2 π 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
2 π ( t 2 A 1 A 0 ) > t P 0 P 1 4 π A 0 > P 0 2 4 π 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
2 π A 1 t 2 P 0 P 1 t + 2 π 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 P t + A 0 ; r t R .
(18)
By the two special cases of inequality (17):
2 π A 1 r 01 2 P 0 P 1 r 01 + 2 π A 0 0 ; 2 π A 1 R 01 2 P 0 P 1 R 01 + 2 π 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 4 π A 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 ) = 2 π A 1 t 2 P 0 P 1 t + 2 π 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 ) = 2 π A 1 t 2 P 0 P 1 t + 2 π 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
2 π A 1 t 2 P 0 P 1 t + 2 π 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, r t R , 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].

Declarations

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).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Southwest University
(2)
School of Mathematical Science, Kaili University
(3)
Faculty of Education, Southwest University

References

  1. Hadwiger H: Die isoperimetrische Ungleichung in Raum. Elem. Math. 1948, 3: 25–38.MathSciNetMATHGoogle Scholar
  2. Hadwiger H: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin; 1957.View ArticleMATHGoogle Scholar
  3. Santaló LA: Integral Geometry and Geometric Probability. Addison-Wesley, Reading; 1976.MATHGoogle Scholar
  4. Zhou J: A kinematic formula and analogous of Hadwiger’s theorem in space. Contemp. Math. 1992, 140: 159–167.View ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.MATHGoogle 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/S1446788700038660View ArticleMathSciNetMATHGoogle 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-XView ArticleMathSciNetMATHGoogle Scholar
  9. Zhou J: On Bonnesen-type inequalities. Acta Math. Sinica (Chin. Ser.) 2007, 50(6):1397–1402.MathSciNetMATHGoogle Scholar
  10. Banchoff TF, Pohl WF: A generalization of the isoperimetric inequality. J. Differ. Geom. 1971, 6: 175–213.MathSciNetMATHGoogle Scholar
  11. Bokowski J, Heil E: Integral representation of quermassintegrals and Bonnesen-style inequalities. Arch. Math. 1986, 47: 79–89. 10.1007/BF01202503MathSciNetView ArticleMATHGoogle Scholar
  12. Burago YD, Zalgaller VA: Geometric Inequalities. Springer, Berlin; 1988.View ArticleMATHGoogle 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/BF02922171MathSciNetView ArticleMATHGoogle Scholar
  14. Croke C: A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv. 1984, 59(2):187–192.MathSciNetView ArticleMATHGoogle 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))MATHGoogle 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.MathSciNetMATHGoogle Scholar
  17. Grinberg, E, Ren, D, Zhou, J: The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature. PreprintGoogle Scholar
  18. Grinberg E: Isoperimetric inequalities and identities for k -dimensional cross-sections of convex bodies. Math. Ann. 1991, 291: 75–86. 10.1007/BF01445191MathSciNetView ArticleMATHGoogle Scholar
  19. Grinberg E, Zhang G: Convolutions, transforms, and convex bodies. Proc. Lond. Math. Soc. 1999, 78: 77–115. 10.1112/S0024611599001653MathSciNetView ArticleMATHGoogle 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-XMathSciNetView ArticleMATHGoogle 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/S0002-9939-98-04336-6View ArticleMathSciNetMATHGoogle Scholar
  22. Hsiang WY: An elementary proof of the isoperimetric problem. Chin. Ann. Math., Ser. A 2002, 23(1):7–12.MathSciNetMATHGoogle Scholar
  23. Hsiung CC: Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary. Ann. Math. 1961, 73(2):213–220.MathSciNetView ArticleMATHGoogle Scholar
  24. Kotlyar BD: On a geometric inequality. Ukr. Geom. Sb. 1987, 30: 49–52.MathSciNetMATHGoogle 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-xMathSciNetView ArticleMATHGoogle 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/s11425-010-4018-3MathSciNetView ArticleMATHGoogle 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/BF01399507MathSciNetView ArticleMATHGoogle Scholar
  28. Lutwak E, Yang D, Zhang G:Sharp affine L p Sobolev inequality. J. Differ. Geom. 2002, 62: 17–38.MathSciNetMATHGoogle 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/S0012-7094-00-10432-2MathSciNetView ArticleMATHGoogle 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/S0002-9904-1978-14553-4MathSciNetView ArticleMATHGoogle Scholar
  32. Osserman R: Bonnesen-style isoperimetric inequality. Am. Math. Mon. 1979, 86: 1–29. 10.2307/2320297MathSciNetView ArticleMATHGoogle Scholar
  33. Pleijel A: On konvexa kurvor. Nord. Mat. Tidskr. 1955, 3: 57–64.MathSciNetGoogle 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.2012.09.004MathSciNetView ArticleMATHGoogle Scholar
  36. Ren D: Topics in Integral Geometry. World Scientific, Singapore; 1994.MATHGoogle 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.View ArticleGoogle 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.1988.134.173MathSciNetView ArticleMATHGoogle Scholar
  39. Schneider R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge; 1993.View ArticleMATHGoogle Scholar
  40. Stone A: On the isoperimetric inequality on a minimal surface. Calc. Var. Partial Differ. Equ. 2003, 17(4):369–391. 10.1007/s00526-002-0174-9View ArticleMathSciNetMATHGoogle Scholar
  41. Tang D: Discrete Wirtinger and isoperimetric type inequalities. Bull. Aust. Math. Soc. 1991, 43: 467–474. 10.1017/S0004972700029312View ArticleMathSciNetMATHGoogle 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/BF01246751MathSciNetView ArticleMATHGoogle Scholar
  44. Teufel E: Isoperimetric inequalities for closed curves in spaces of constant curvature. Results Math. 1992, 22: 622–630. 10.1007/BF03323109MathSciNetView ArticleMATHGoogle Scholar
  45. Wei S, Zhu M: Sharp isoperimetric inequalities and sphere theorems. Pac. J. Math. 2005, 220(1):183–195. 10.2140/pjm.2005.220.183MathSciNetView ArticleMATHGoogle 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.1975.24.24021MathSciNetView ArticleMATHGoogle Scholar
  47. Weiner JL: Isoperimetric inequalities for immersed closed spherical curves. Proc. Am. Math. Soc. 1994, 120(2):501–506. 10.1090/S0002-9939-1994-1163337-4MathSciNetView ArticleMATHGoogle Scholar
  48. Yau ST: Isoperimetric constants and the first eigenvalue of a compact manifold. Ann. Sci. Éc. Norm. Super. 1975, 8(4):487–507.MathSciNetMATHGoogle 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.View ArticleGoogle Scholar
  51. Zhang X-M: Bonnesen-style inequalities and pseudo-perimeters for polygons. J. Geom. 1997, 60: 188–201. 10.1007/BF01252226MathSciNetView ArticleMATHGoogle Scholar
  52. Zhang X-M: Schur-convex functions and isoperimetric inequalities. Proc. Am. Math. Soc. 1998, 126(2):461–470. 10.1090/S0002-9939-98-04151-3View ArticleMathSciNetMATHGoogle Scholar
  53. Zhao L, Ma L, Zhou J: A geometric application of the Wirtinger inequality. J. Math. (Wuhan) 2011, 31(5):887–890.MathSciNetMATHGoogle Scholar
  54. Zhou J: On Willmore inequality for submanifolds. Can. Math. Bull. 2007, 50(3):474–480. 10.4153/CMB-2007-047-4View ArticleMATHGoogle 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.MathSciNetView ArticleMATHGoogle 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/s11425-007-0029-0MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.2013.50.1.175MathSciNetView ArticleMATHGoogle Scholar
  61. Bonnesen T: Les probléms des isopérimétres et des isépiphanes. Gauthier-Villars, Paris; 1929.MATHGoogle 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.View ArticleGoogle 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.MathSciNetMATHGoogle 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.MathSciNetView ArticleMATHGoogle Scholar
  65. Fusco N, Maggi F, Pratelli A: The sharp quantitative isoperimetric inequality. Ann. Math. 2008, 168(3):941–980. 10.4007/annals.2008.168.941MathSciNetView ArticleMATHGoogle 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-zMathSciNetView ArticleMATHGoogle 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.View ArticleMATHGoogle Scholar
  69. Bonnesen T, Fenchel W: Theorie der konvexen Köeper. 2nd edition. Springer, Berlin; 1974.View ArticleMATHGoogle Scholar
  70. Bottema O: Eine obere Grenze für das isoperimetrische Defizit ebener Kurven. Nederl. Akad. Wetensch. Proc. 1933, A66: 442–446.MATHGoogle Scholar
  71. Flanders H: A proof of Minkowski’s inequality for convex curves. Am. Math. Mon. 1968, 75: 581–593. 10.2307/2313773MathSciNetView ArticleMATHGoogle Scholar
  72. Hardy G, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1951.MATHGoogle Scholar
  73. Howard R: Blaschke’s rolling theorem for manifolds with boundary. Manuscr. Math. 1999, 99(4):471–483. 10.1007/s002290050186View ArticleMATHGoogle Scholar
  74. Koutrouieiotis D: On Blaschke’s rolling theorems. Arch. Math. 1972, 21(1):655–660.View ArticleMATHGoogle Scholar
  75. Zhang G: The affine Sobolev inequality. J. Differ. Geom. 1999, 53: 183–202.MathSciNetMATHGoogle Scholar
  76. Zhang G: Geometric inequalities and inclusion measures of convex bodies. Mathematika 1994, 41: 95–116. 10.1112/S0025579300007208MathSciNetView ArticleMATHGoogle Scholar
  77. Zhang, G: Convex Geometric Analysis. PreprintGoogle Scholar
  78. Zhou J, Xia Y, Zeng C: Some new Bonnesen-style inequalities. J. Korean Math. Soc. 2011, 48(2):421–430. 10.4134/JKMS.2011.48.2.421MathSciNetView ArticleMATHGoogle 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/s10114-012-9657-6MathSciNetView ArticleMATHGoogle 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.MathSciNetGoogle Scholar

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