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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2014 | Research

Some Bonnesen-style Minkowski inequalities

verfasst von: Wenxue Xu, Chunna Zeng, Jiazu Zhou

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2014

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Abstract

In this paper, we obtain some Bonnesen-style Minkowski inequalities of mixed volumes of convex bodies K and L in the Euclidean space

R^{n}

. Let L be the unit ball; we get some better Bonnesen-style isoperimetric inequalities than Dinghas’s result for

n \geq 3

MSC:52A20, 52A40.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

1 Introduction

It is well known that the ball has the maximum volume among bodies of fixed surface area in the Euclidean space

R^{n}

. That is, of all domains K with surface area

S (K)

and volume

V (K)

(cf. [1, 2]),

S {(K)}^{n} - n^{n} ω_{n} V {(K)}^{n - 1} \geq 0,

(1)

with equality if and only if K is a ball. Here

ω_{n}

denotes the volume of the unit ball,

ω_{n} = \frac{2 π^{n / 2}}{n Γ (n / 2)},

where

Γ (\cdot)

is the Gamma function.

The isoperimetric deficit

Δ_{n} (K) = S {(K)}^{n} - n^{n} ω_{n} V {(K)}^{n - 1}

(2)

measures the deficit between the domain K and a ball of radius

{(S (K) / n ω_{n})}^{1 / (n - 1)}

. A Bonnesen-style isoperimetric inequality is of the form (cf. [2‐4])

Δ_{n} (K) = S {(K)}^{n} - n^{n} ω_{n} V {(K)}^{n - 1} \geq B_{K},

(3)

where the quantity

B_{K}

is a non-negative invariant of geometric significance of K and vanishes only when K is a ball.

Bonnesen himself proved several inequalities of the form (3) in the Euclidean plane (cf. [5, 6]), but he was not able to obtain direct generalizations of his two-dimensional results. This was done much later, first by Hadwiger [7] for

n = 3

, and then by Dinghas [8] for arbitrary dimension. From then on, some Bonnesen-style inequalities in the higher dimensions and generalizations have been obtained by Osserman (cf. [1, 2]), Santaló (cf. [9]), Groemer and Schneider (cf. [10]), Zhang (cf. [11]), Zhou (cf. [4, 12]) and others. See references [13‐36] for more details. The following well-known Bonnesen-style inequality for a convex body K in the Euclidean space

R^{n}

is due to Dinghas (cf. [8]):

S {(K)}^{n} - n^{n} ω_{n} V {(K)}^{n - 1} \geq {(S {(K)}^{1 / (n - 1)} - {(n ω_{n})}^{1 / (n - 1)} r)}^{n (n - 1)},

(4)

where r is the in-radius of K, and equality holds if and only if K is a ball.

In [11], some different forms of Bonnesen-style isoperimetric inequalities have been established associated with the mean width of K. Zhang obtained (cf. [11])

{(\frac{M (K)}{2})}^{n / (n - 1)} - {(\frac{V (K)}{ω_{n}})}^{1 / (n - 1)} \geq {(\frac{V (K)}{ω_{n}})}^{n / (n - 1)} ({(\frac{V (K)}{ω_{n}})}^{- 1 / n} - R^{- 1}),

where

M (K)

and R are the mean width and out-radius of K, respectively.

The Minkowski inequality of mixed volume is a natural generalization of the isoperimetric inequality (1) in the Euclidean space

R^{n}

(cf. [27, 37‐39]). Let K, L be convex bodies in

R^{n}

, then

V_{1} {(K, L)}^{n} \geq V {(K)}^{n - 1} V (L),

(5)

where

V_{1} (K, L)

is the mixed volume of K and L and the equality holds if and only if K and L are homothetic.

Motivated by (2), we define the Minkowski homothetic deficit as

Δ_{n} (K, L) = V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L) .

(6)

The Minkowski homothetic deficit

Δ_{n} (K, L)

measures the homothety between K and L. Then a Bonnesen-style Minkowski inequality would be of the form

Δ_{n} (K, L) = V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L) \geq B_{K, L},

(7)

where the quantity

B_{K, L}

is an invariant of geometric significance about K and L with the following basic properties:

B_{K, L}

is non-negative;

B_{K, L}

vanishes only when K and L are homothetic.

Note that let L be the unit ball B and by

S (K) = n V_{1} (K, B)

, the surface area of K, then the Minkowski homothetic deficit is just the isoperimetric deficit. Therefore, the Bonnesen-style Minkowski inequality (7) is more general than the Bonnesen-style isoperimetric inequality (3).

In this paper, we focus on Bonnesen-style Minkowski inequalities of type (7). Some

B_{K, L}

are obtained. Let L be the unit ball; then we obtain stronger Bonnesen-style isoperimetric inequalities K than (4).

2 Preliminaries

A set of points K in the Euclidean space

R^{n}

is convex if for all

x, y \in K

and

0 \leq λ \leq 1

λ x + (1 - λ) y \in K

. A domain is a set with nonempty interiors. A convex body is a compact convex domain. The set of convex bodies in

R^{n}

is denoted by

K^{n}

. Let

K_{o}^{n}

be the class of members of

K^{n}

containing the origin in their interiors. Write V for an n-dimensional Lebesgue measure and

H^{n - 1}

for an

(n - 1)

-dimensional Hausdorff measure.

S^{n - 1}

denotes the surface of the unit ball in

R^{n}

A convex body

K \subset R^{n}

is uniquely determined by its support function

h_{K} : R^{n} \to R

, where

h_{K} (x) = max {x \cdot y : y \in K}

, for

x \in R^{n}

. For the support function of the dilate

c K = {c x : x \in K}

of a convex body K we have

h_{c K} = c h_{K}, c > 0 .

(8)

Note that support functions are positively homogeneous of degree one and subadditive. It follows immediately from the definition of support functions that for convex bodies K and L

K \subseteq L ⟺ h_{K} \leq h_{L} .

(9)

For a convex body K and each Borel set

ω \subset S^{n - 1}

, the reverse spherical image

τ (K, ω)

, of K at ω is the set of all boundary points of K which have an outer unit normal belonging to the set ω. Associated with each convex body

K \in K_{o}^{n}

there is a Borel measure

S_{K}

S^{n - 1}

called the Aleksandrov-Fenchel surface area measure of K, defined by

S_{K} (ω) = H^{n - 1} (τ (K, ω)),

for each Borel set

ω \subseteq S^{n - 1}

. Observe that for the surface area measure of the dilate cK of K we have

S_{c K} = c^{n - 1} S_{K}, c > 0 .

The Minkowski sum of convex sets

K_{1}, \dots, K_{m}

R^{n}

is defined by

K_{1} + \dots + K_{m} = {x_{1} + \dots + x_{m} : x_{1} \in K_{1}, \dots, x_{m} \in K_{m}} .

The mixed volume

V (K_{1}, \dots, K_{n})

of compact convex sets

K_{1}, \dots, K_{n}

R^{n}

is defined by

V (K_{1}, \dots, K_{n}) = \frac{1}{n!} \sum_{j = 1}^{n} {(- 1)}^{n + j} \sum_{i_{1} < \dots < i_{k}} V (K_{i_{1}} + \dots + K_{i_{k}}) .

The Aleksandrov-Fenchel inequality about the i th mixed volume is

V_{i} {(K_{1}, K_{2})}^{2} \geq V_{i + 1} (K_{1}, K_{2}) V_{i - 1} (K_{1}, K_{2}),

(10)

where

V_{i} (K_{1}, K_{2}) = V (\underset{n - i}{\underset{⏟}{K_{1}, \dots, K_{1}}}, \underset{i}{\underset{⏟}{K_{2}, \dots, K_{2}}})

with

K_{1}

appears

n - i

times and

K_{2}

appears i times and (10) holds as an equality if and only if K and L are homothetic.

Note that

V_{n} (K_{1}, K_{2}) = V (K_{2}), V_{0} (K_{1}, K_{2}) = V (K_{1}) .

(11)

The following inequality for mixed volumes is the general Aleksandrov-Fenchel inequality: Let

K_{1}, \dots, K_{n} \in K

and

1 \leq m \leq n

. Then

V {(K_{1}, \dots, K_{n})}^{m} \geq \prod_{i = 1}^{m} V (K_{i}, \dots, K_{i}, K_{m + 1}, \dots, K_{n}) .

Hence

V_{1} {(K_{1}, K_{2})}^{n - 1} \geq V {(K_{1})}^{n - 2} V_{n - 1} (K_{1}, K_{2}) .

(12)

Let

K_{2} = B

, then

V_{i} (K_{1}, B) = W_{i} (K_{1})

, the i th quermassintegral of the convex body

K_{1}

The mixed volume has monotonicity: If

K_{1} \subset K_{1}^{'}

, then

V (K_{1}, K_{2}, \dots, K_{n}) \leq V (K_{1}^{'}, K_{2}, \dots, K_{n}) .

The mixed volume

V_{1} (K, L)

of the convex bodies

K, L \in K_{o}^{n}

has the integral form

V_{1} (K, L) = \frac{1}{n} \int_{S^{n - 1}} h_{L} d S_{K} .

(13)

Since

V (K) = V_{1} (K, K),

we have

V (K) = \frac{1}{n} \int_{S^{n - 1}} h_{K} d S_{K} .

If B is the unit ball, then

n V_{1} (K, B) = S (K),

the surface area of K. The mean width

M (K)

of K is

M (K) = \frac{2}{ω_{n}} V_{1} (B, K),

that is,

M (K) = \frac{2}{n ω_{n}} \int_{S^{n - 1}} h_{K} d S_{K} .

The in-radius

r (K, L)

, out-radius

R (K, L)

of K with respect to L are, respectively, defined by

\begin{matrix} r (K, L) = sup {λ : x \in R^{n} and x + λ L \subset K}, \\ R (K, L) = inf {λ : x \in R^{n} and K \subset x + λ L} . \end{matrix}

Notice that always

r (K, L) R (L, K) = 1 .

When L is the unit ball,

r (K, L)

and

R (K, L)

are the radius of maximal inscribed and minimal circumscribed balls of K, respectively.

3 Bonnesen-style Minkowski inequalities associated with $r (K, L)$

In this section, we derive some Bonnesen-style Minkowski inequalities associated with in-radius

r (K, L)

of K with respect to L. In [26], Diskant improved the Minkowski inequality of mixed volumes as follows.

Lemma 1 Let K, L be convex bodies in the Euclidean space

R^{n}

, then

V_{1} {(K, L)}^{n / (n - 1)} - V (K) V {(L)}^{1 / (n - 1)} \geq {(V_{1} {(K, L)}^{1 / (n - 1)} - r (K, L) V {(L)}^{1 / (n - 1)})}^{n},

(14)

with equality if and only if K is homothetic to L.

Note that the right-hand side of (14) is non-negative for

x + r (K, L) L \subseteq K

(

x \in R^{n}

). By (13) we have

V_{1} (K, L) = \frac{1}{n} \int_{S^{n - 1}} h_{L} d S_{K} \geq \frac{1}{n} \int_{S^{n - 1}} h_{L} d S_{r (K, L) L} \geq r {(K, L)}^{n - 1} V (L) .

From Lemma 1 and using the inequality

x^{n - 1} - y^{n - 1} \geq {(x - y)}^{n - 1}

(for

x \geq y \geq 0

), a lower bound of the Minkowski deficit follows (cf. [26, 27]).

Proposition 1 Let K, L be convex bodies in the Euclidean space

R^{n}

, then

Δ_{n} (K, L) \geq {(V_{1} {(K, L)}^{1 / (n - 1)} - r (K, L) V {(L)}^{1 / (n - 1)})}^{n (n - 1)},

(15)

where the inequality holds as an equality if and only if K and L are homothetic.

The following Bonnesen-style Minkowski inequality is stronger than (15) for

n = 3

Theorem 1 Let K, L be convex bodies in the Euclidean space

R^{3}

, then

\begin{array}{rcl} Δ_{3} (K, L) & \geq & {(V_{1} {(K, L)}^{1 / 2} - r (K, L) V {(L)}^{1 / 2})}^{6} \\ + 2 r {(K, L)}^{3} V {(L)}^{3 / 2} {(V_{1} {(K, L)}^{1 / 2} - r (K, L) V {(L)}^{1 / 2})}^{3}, \end{array}

(16)

with equality if and only if K is homothetic to L.

Proof Since

V (K) \geq r {(K, L)}^{3} V (L)

and by

x^{3} - y^{3} \geq {(x - y)}^{3}

(for

x \geq y \geq 0

), we have

\begin{array}{rcl} V_{1} {(K, L)}^{3 / 2} + V (K) V {(L)}^{1 / 2} & \geq & V_{1} {(K, L)}^{3 / 2} + r {(K, L)}^{3} V {(L)}^{3 / 2} \\ = & V_{1} {(K, L)}^{3 / 2} - {(r (K, L) V {(L)}^{1 / 2})}^{3} + 2 r {(K, L)}^{3} V {(L)}^{3 / 2} \\ \geq & {(V_{1} {(K, L)}^{1 / 2} - r (K, L) V {(L)}^{1 / 2})}^{3} + 2 r {(K, L)}^{3} V {(L)}^{3 / 2} . \end{array}

Note that (14) can be rewritten as

V_{1} {(K, L)}^{3 / 2} - V (K) V {(L)}^{1 / 2} \geq {(V_{1} {(K, L)}^{1 / 2} - r (K, L) V {(L)}^{1 / 2})}^{3} .

Multiplying by

V_{1} {(K, L)}^{3 / 2} + V (K) V {(L)}^{1 / 2}

on both sides, we have

V_{1} {(K, L)}^{3} - V {(K)}^{2} V (L) \geq {(V_{1} {(K, L)}^{1 / 2} - r (K, L) V {(L)}^{1 / 2})}^{3} (V_{1} {(K, L)}^{3 / 2} + V (K) V {(L)}^{1 / 2}) .

By these inequalities, we complete the proof of the theorem. □

Let L be the unit ball and notice

S (K) = 3 V_{1} (K, B)

in (16), we obtain the following Bonnesen-style isoperimetric inequality that strengthens Dinghas’s inequality (4) for

n = 3

Corollary 1 Let K be a convex body in

R^{3}

and r be the in-radius of K, then

Δ_{3} (K) \geq {(S {(K)}^{1 / 2} - {(4 π)}^{1 / 2} r)}^{6} + 16 π^{3 / 2} r^{3} {(S {(K)}^{1 / 2} - {(4 π)}^{1 / 2} r)}^{3},

(17)

with equality if and only if K is a ball.

For

n \geq 4

, we obtain a stronger Bonnesen-style Minkowski inequality as follows.

Theorem 2 Let K, L be convex bodies in the Euclidean space

R^{n}

(

n \geq 4

), then

\begin{array}{rcl} Δ_{n} (K, L) & \geq & {(V_{1} {(K, L)}^{1 / (n - 1)} - r (K, L) V {(L)}^{1 / (n - 1)})}^{n (n - 1)} \\ + 2 {(r (K, L) V {(L)}^{1 / (n - 1)})}^{n (n - 2)} {(V_{1} {(K, L)}^{1 / (n - 1)} - r (K, L) V {(L)}^{1 / (n - 1)})}^{n} \\ + {(V_{1} (K, L) V (L))}^{n / (n - 1)} r {(K, L)}^{n} {(V_{1} {(K, L)}^{1 / (n - 1)} - r (K, L) V {(L)}^{1 / (n - 1)})}^{n (n - 3)}, \end{array}

with equality if and only if K is homothetic to L.

Proof Let

p = V_{1} {(K, L)}^{n / (n - 1)}

and

q = V {(L)}^{n / (n - 1)} r {(K, L)}^{n}

, then

p \geq q

\begin{matrix} \sum_{i = 2}^{n} (V_{1} {(K, L)}^{n (n - i) / (n - 1)} {(V {(K)}^{n - 1} V (L))}^{(i - 2) / (n - 1)}) \\ \geq \sum_{i = 2}^{n} (V_{1} {(K, L)}^{n (n - i) / (n - 1)} {(V {(L)}^{1 / (n - 1)} r (K, L))}^{n (i - 2)}) \\ = (p^{n - 2} - q^{n - 2}) + p q (p^{n - 4} + p^{n - 5} q + \dots + p^{n - i} q^{i - 4} + \dots + q^{n - 4}) + 2 q^{n - 2} \\ = (p^{n - 2} - q^{n - 2}) + p q \cdot \frac{p^{n - 3} - q^{n - 3}}{p - q} + 2 q^{n - 2} \\ \geq {(p - q)}^{n - 2} + p q {(p - q)}^{n - 4} + 2 q^{n - 2} \\ \geq {(p^{1 / n} - q^{1 / n})}^{n (n - 2)} + p q {(p^{1 / n} - q^{1 / n})}^{n (n - 4)} + 2 q^{n - 2} . \end{matrix}

That is,

\begin{matrix} \sum_{i = 2}^{n} (V_{1} {(K, L)}^{n (n - i) / (n - 1)} {(V {(K)}^{n - 1} V (L))}^{(i - 2) / (n - 1)}) \\ \geq {(V_{1} {(K, L)}^{1 / (n - 1)} - V {(L)}^{1 / (n - 1)} r (K, L))}^{n (n - 2)} + 2 {(V {(L)}^{1 / (n - 1)} r (K, L))}^{n (n - 2)} \\ + {(V_{1} (K, L) V (L))}^{n / (n - 1)} r {(K, L)}^{n} {(V_{1} {(K, L)}^{1 / (n - 1)} - V {(L)}^{1 / (n - 1)} r (K, L))}^{n (n - 4)} . \end{matrix}

Multiplying by

\sum_{i = 2}^{n} (V_{1} {(K, L)}^{n (n - i) / n - 1} {(V {(K)}^{n - 1} V (L))}^{(i - 2) / (n - 1)})

both sides of (14) and via the formula

a^{n - 1} - b^{n - 1} = (a - b) (a^{n - 2} + a^{n - 3} b + \dots + a^{n - i} b^{i - 2} + \dots + a b^{n - 3} + b^{n - 2}),

we obtain

\begin{array}{rcl} V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L) & \geq & {(V_{1} {(K, L)}^{1 / (n - 1)} - r (K, L) V {(L)}^{1 / (n - 1)})}^{n} \\ \times (\sum_{i = 2}^{n} (V_{1} {(K, L)}^{n (n - i) / (n - 1)} {(V {(K)}^{n - 1} V (L))}^{(i - 2) / (n - 1)})) . \end{array}

We complete the proof of Theorem 2. □

Let L be the unit ball and by

S (K) = n V_{1} (K, B)

in Theorem 2; we obtain the following stronger Bonnesen-style isoperimetric inequality than Dinghas’s inequality (4) for

n \geq 4

Corollary 2 Let K be a convex body in

R^{n}

(

n \geq 4

) and r be the in-radius of K, then

\begin{array}{rcl} Δ_{n} (K) & \geq & {(S {(K)}^{1 / (n - 1)} - {(n ω_{n})}^{1 / (n - 1)} r)}^{n (n - 1)} \\ + 2 {({(n ω_{n})}^{1 / (n - 1)} r)}^{n (n - 2)} {(S {(K)}^{1 / (n - 1)} - {(n ω_{n})}^{1 / (n - 1)} r)}^{n} \\ + {(n ω_{n} S (K))}^{n / (n - 1)} r^{n} {(S {(K)}^{1 / (n - 1)} - {(n ω_{n})}^{1 / (n - 1)} r)}^{n (n - 3)}, \end{array}

with equality if and only if K is a ball.

4 Bonnesen-style Minkowski inequalities associated with the mean width

In this section, we derive some Bonnesen-style Minkowski inequalities associated with the mean width.

Lemma 2 Let K, L be convex bodies in

R^{n}

, then

\frac{V_{0} (K, L)}{V_{1} (K, L)} \leq {(\frac{V_{0} (K, L)}{V_{n} (K, L)})}^{1 / n} \leq \frac{V_{n - 1} (K, L)}{V_{n} (K, L)} \leq \frac{V_{1} {(K, L)}^{n - 1}}{V {(K)}^{n - 2} V (L)},

(18)

with equality if and only if K and L are homothetic.

Proof By inequality (12), we have

\frac{V_{n - 1} (K, L)}{V (L)} \leq \frac{V_{1} {(K, L)}^{n - 1}}{V {(K)}^{n - 2} V (L)} .

By the Aleksandrov-Fenchel inequality (10) we have

\frac{V_{0} (K, L)}{V_{1} (K, L)} \leq \frac{V_{1} (K, L)}{V_{2} (K, L)} \leq \dots \leq \frac{V_{i} (K, L)}{V_{i + 1} (K, L)} \leq \dots \leq \frac{V_{n - 1} (K, L)}{V_{n} (K, L)} .

Therefore

\frac{V_{0} (K, L)}{V_{1} (K, L)} \leq {(\frac{V_{0} (K, L)}{V_{n} (K, L)})}^{1 / n} \leq \frac{V_{n - 1} (K, L)}{V_{n} (K, L)} .

□

Theorem 3 Let K, L be convex bodies in

R^{n}

, then

Δ_{n} (K, L) \geq V {(K)}^{n - 2} V (L) V_{1} (K, L) (\frac{V_{n - 1} (K, L)}{V_{n} (K, L)} - {(\frac{V_{0} (K, L)}{V_{n} (K, L)})}^{1 / n}),

(19)

with equality if and only if K and L are homothetic.

Proof Via (18), we have

\frac{V_{1} {(K, L)}^{n - 1}}{V {(K)}^{n - 2} V (L)} - \frac{V_{0} (K, L)}{V_{1} (K, L)} \geq \frac{V_{n - 1} (K, L)}{V_{n} (K, L)} - {(\frac{V_{0} (K, L)}{V_{n} (K, L)})}^{1 / n} .

That is

V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L) \geq V {(K)}^{n - 2} V (L) V_{1} (K, L) (\frac{V_{n - 1} (K, L)}{V_{n} (K, L)} - {(\frac{V_{0} (K, L)}{V_{n} (K, L)})}^{1 / n}) .

□

The following Bonnesen-style inequality is a direct consequence of Theorem 3.

Theorem 4 Let K be a convex body in

R^{n}

, then

Δ_{n} (K) \geq n^{n - 1} ω_{n} S (K) V {(K)}^{n - 2} (\frac{M (K)}{2} - {(\frac{V (K)}{ω_{n}})}^{1 / n}),

with equality if and only if K is a ball.

Lemma 3 Let K, L be convex bodies in

R^{n}

, then

\begin{matrix} \frac{V_{1} {(K, L)}^{n - 1} - \sqrt{V_{1} {(K, L)}^{n - 2} (V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L))}}{V {(K)}^{n - 2} V (L)} \\ \leq \frac{V_{0} (K, L)}{V_{1} (K, L)} \leq \frac{V_{n - 1} (K, L)}{V_{n} (K, L)} \leq \frac{V_{1} {(K, L)}^{n - 1}}{V {(K)}^{n - 2} V (L)} . \end{matrix}

(20)

Proof The Minkowski inequality (5) gives

\frac{V_{1} {(K, L)}^{n - 1} - \sqrt{V_{1} {(K, L)}^{n - 2} (V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L))}}{V {(K)}^{n - 2} V (L)} \leq \frac{V_{0} (K, L)}{V_{1} (K, L)} .

The above inequality together with (18) leads to Lemma 3. □

We are now in a position to prove the following Bonnesen-style Minkowski inequality.

Theorem 5 Let K, L be convex bodies in

R^{n}

, then

Δ_{n} (K, L) \geq \frac{V {(K)}^{2 n - 4} V {(L)}^{2}}{V_{1} {(K, L)}^{n - 2}} {(\frac{V_{n - 1} (K, L)}{V_{n} (K, L)} - \frac{V_{0} (K, L)}{V_{1} (K, L)})}^{2},

(21)

with equality if and only if K and L are homothetic.

Proof From (20) we have

\frac{\sqrt{V_{1} {(K, L)}^{n - 2} (V_{1} {(K, L)}^{n} - V {(K)}^{n - 1} V (L))}}{V {(K)}^{n - 2} V (L)} \geq \frac{V_{n - 1} (K, L)}{V_{n} (K, L)} - \frac{V_{0} (K, L)}{V_{1} (K, L)} .

□

The following Bonnesen-style inequality is a direct consequence of Theorem 5 when L is the unit ball.

Theorem 6 Let K be a convex body in

R^{n}

, then

S {(K)}^{n} - n^{n} ω_{n} V {(K)}^{n - 1} \geq \frac{n^{2 n - 2} ω_{n}^{2} V^{2 n - 4}}{S^{n - 2}} {(\frac{M (K)}{2} - \frac{n V (K)}{S (K)})}^{2},

with equality if and only if K is a ball.

Acknowledgements

The 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 authors are supported in part by NSFC (No. 11271302 and No. 11326073), Fundamental Research Funds for the Central Universities (No. XDJK2014C164), the Ph.D. Program of Higher Education Research Funds (No. 2012182110020).

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

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

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

Zhang X-M: Bonnesen-style inequalities and pseudo-perimeters for polygons. J. Geom. 1997, 60: 188-201. 10.1007/BF01252226MathSciNetCrossRefMATH

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

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

Bonnesen T, Fenchel W: Theorie der konvexen. Köerper, Berlin; 1934. 2nd edn. 1974 2nd edn. 1974CrossRefMATH

Hadwiger H: Die isoperimetrische Ungleichung im Raum. Elem. Math. 1948, 3: 25-38.MathSciNetMATH

Dinghas A: Bemerkung zu einer Verschärfung der isoperimetrischen Ungleichung durch H. Hadwiger. Math. Nachr. 1948, 1: 284-286. 10.1002/mana.19480010503MathSciNetCrossRefMATH

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

10.

Groemer H, Schneider R: Stability estimates for some geometric inequalities. Bull. Lond. Math. Soc. 1991, 23: 67-74. 10.1112/blms/23.1.67MathSciNetCrossRefMATH

11.

Zhang G: Geometric inequalities and inclusion measures of convex bodies. Mathematika 1994, 41: 95-116. 10.1112/S0025579300007208MathSciNetCrossRefMATH

12.

Zhou J, Ren D: Geometric inequalities from the viewpoint of integral geometry. Acta Math. Sci. Ser. A Chin. Ed. 2010, 30: 1322-1339.MathSciNetMATH

13.

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

14.

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

15.

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

16.

Bokowski J, Heil E: Integral representation of quermassintegrals and Bonnesen-style inequalities. Arch. Math. 1986, 47: 79-89. 10.1007/BF01202503MathSciNetCrossRefMATH

17.

Burago YuD, Zalgaller VA: Geometric Inequalities. Springer, Berlin; 1988.CrossRefMATH

18.

Gage M: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 1983,50(4):1225-1229. 10.1215/S0012-7094-83-05052-4MathSciNetCrossRefMATH

19.

Green M, Osher S: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J. Math. 1999,3(3):659-676.MathSciNetCrossRefMATH

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-XMathSciNetCrossRefMATH

21.

Hsiang WY: An elementary proof of the isoperimetric problem. Chin. Ann. Math. 2002,23(1):7-12.MathSciNetMATH

22.

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

23.

Sangwine-Yager JR: Bonnesen-style inequalities for Minkowski relative geometry. Trans. Am. Math. Soc. 1988,307(1):373-382. 10.1090/S0002-9947-1988-0936821-5MathSciNetCrossRefMATH

24.

Diskant V: Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference. Sib. Mat. Zh. 1972 (in Russian), English translation: Siberian Math. J. 13, 529-532 (1973), 13: 767-772. (in Russian), English translation: Siberian Math. J. 13, 529-532 (1973)MathSciNet

25.

Diskant V: Strengthening of an isoperimetric inequality. Sib. Mat. Zh. 1973 (in Russian), English translation: Siberian Math. J. 14, 608-611 (1973), 14: 873-877. (in Russian), English translation: Siberian Math. J. 14, 608-611 (1973)MathSciNet

26.

Diskant V: A generalization of Bonnesen’s inequalities. Sov. Math. Dokl. 1973, 14: 1728-1731.MATH

27.

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

28.

Hadwiger H: Kurze Herleitung einer verschärften isoperimetrischen Ungleichung für konvexe Körper. Revue Fac. Sci. Univ. Istanbul, Sér. A 1949, 14: 1-6.MathSciNetMATH

29.

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

30.

Klain D: Bonnesen-type inequalities for surfaces of constant curvature. Adv. Appl. Math. 2007,39(2):143-154. 10.1016/j.aam.2006.11.004MathSciNetCrossRefMATH

31.

Ren D: Topics in Integral Geometry. World Scientific, Sigapore; 1994.MATH

32.

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

33.

Gao X: A new reverse isoperimetric inequality and its stability. Math. Inequal. Appl. 2012,15(3):733-743.MathSciNetMATH

34.

Pan S, Xu H: Stability of a reverse isoperimetric inequality. J. Math. Anal. Appl. 2009, 350: 348-353. 10.1016/j.jmaa.2008.09.047MathSciNetCrossRefMATH

35.

Pan S, Tang X, Wang X: A refined reverse isoperimetric inequality in the plane. Math. Inequal. Appl. 2010,13(2):329-338.MathSciNetMATH

36.

Xu W, Zhou J, Zhu B: On containment measure and the mixed isoperimetric inequality. J. Inequal. Appl. 2013. Article ID 540, 2013: Article ID 540

37.

Gardner R: Geometric Tomography. Cambridge University Press, New York; 1995.MATH

38.

Gardner R: The Brunn-Minkowski inequality. Bull. Am. Math. Soc. 2002,39(3):355-405. 10.1090/S0273-0979-02-00941-2CrossRefMathSciNetMATH

39.

Gardner R: The Brunn-Minkowski inequality, Minkowski’s first inequality, and their duals. J. Math. Anal. Appl. 2000, 245: 502-512. 10.1006/jmaa.2000.6774MathSciNetCrossRefMATH

Titel: Some Bonnesen-style Minkowski inequalities
verfasst von: Wenxue Xu
Chunna Zeng
Jiazu Zhou
Publikationsdatum: 01.12.2014
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2014
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2014-270

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Some Bonnesen-style Minkowski inequalities

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

3 Bonnesen-style Minkowski inequalities associated with $r (K, L)$

4 Bonnesen-style Minkowski inequalities associated with the mean width

Acknowledgements

Competing interests

Authors’ contributions

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Springer Professional

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

3 Bonnesen-style Minkowski inequalities associated with r ( K , L )

4 Bonnesen-style Minkowski inequalities associated with the mean width

Acknowledgements

Competing interests

Authors’ contributions

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3 Bonnesen-style Minkowski inequalities associated with $r (K, L)$