Some Bonnesen-style Minkowski inequalities

Abstract

In this paper, we obtain some Bonnesen-style Minkowski inequalities of mixed volumes of convex bodies K and L in the Euclidean space ${\mathbb{R}}^n$. Let L be the unit ball; we get some better Bonnesen-style isoperimetric inequalities than Dinghas’s result for $n\ge 3$.

MSC:52A20, 52A40.

1 Introduction

It is well known that the ball has the maximum volume among bodies of fixed surface area in the Euclidean space ${\mathbb{R}}^n$. That is, of all domains K with surface area $S(K)$ and volume $V(K)$ (cf. [1, 2]),

$$S{(K)}^n-n^n{\omega }_{n}V{(K)}^{n-1}\ge 0,$$

(1)

with equality if and only if K is a ball. Here ${\omega }_n$ denotes the volume of the unit ball,

$${\omega }_n=\frac{2{\pi }^{n/2}}{n\mathrm{\Gamma }(n/2)},$$

where $\mathrm{\Gamma }(\cdot )$ is the Gamma function.

The isoperimetric deficit

$${\mathrm{\Delta }}_n(K)=S{(K)}^n-n^n{\omega }_{n}V{(K)}^{n-1}$$

(2)

measures the deficit between the domain K and a ball of radius ${(S(K)/n{\omega }_n)}^{1/(n-1)}$. A Bonnesen-style isoperimetric inequality is of the form (cf. [2–4])

$${\mathrm{\Delta }}_n(K)=S{(K)}^n-n^n{\omega }_{n}V{(K)}^{n-1}\ge B_K,$$

(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 ${\mathbb{R}}^n$ is due to Dinghas (cf. [8]):

$$S{(K)}^n-n^n{\omega }_{n}V{(K)}^{n-1}\ge {(S{(K)}^{1/(n-1)}-{(n{\omega }_n)}^{1/(n-1)}r)}^{n(n-1)},$$

(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])

$${\left(\frac{M(K)}{2}\right)}^{n/(n-1)}-{\left(\frac{V(K)}{{\omega }_n}\right)}^{1/(n-1)}\ge {\left(\frac{V(K)}{{\omega }_n}\right)}^{n/(n-1)}({\left(\frac{V(K)}{{\omega }_n}\right)}^{-1/n}-R^{-1}),$$

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 ${\mathbb{R}}^n$ (cf. [27, 37–39]). Let K, L be convex bodies in ${\mathbb{R}}^n$, then

$$V_1{(K,L)}^n\ge V{(K)}^{n-1}V(L),$$

(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

$${\mathrm{\Delta }}_n(K,L)=V_1{(K,L)}^n-V{(K)}^{n-1}V(L).$$

(6)

The Minkowski homothetic deficit ${\mathrm{\Delta }}_n(K,L)$ measures the homothety between K and L. Then a Bonnesen-style Minkowski inequality would be of the form

$${\mathrm{\Delta }}_n(K,L)=V_1{(K,L)}^n-V{(K)}^{n-1}V(L)\ge B_{K,L},$$

(7)

where the quantity $B_{K,L}$ is an invariant of geometric significance about K and L with the following basic properties:

• 1. $B_{K,L}$ is non-negative;

• 2. $B_{K,L}$ 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 ${\mathbb{R}}^n$ is convex if for all $x,y\in K$ and $0\le \lambda \le 1$, $\lambda x+(1-\lambda )y\in K$. A domain is a set with nonempty interiors. A convex body is a compact convex domain. The set of convex bodies in ${\mathbb{R}}^n$ is denoted by ${\mathcal{K}}^n$. Let ${\mathcal{K}}_o^n$ be the class of members of ${\mathcal{K}}^n$ containing the origin in their interiors. Write V for an n-dimensional Lebesgue measure and ${\mathcal{H}}^{n-1}$ for an $(n-1)$-dimensional Hausdorff measure. $S^{n-1}$ denotes the surface of the unit ball in ${\mathbb{R}}^n$.

A convex body $K\subset {\mathbb{R}}^n$ is uniquely determined by its support function $h_K:{\mathbb{R}}^n\to \mathbb{R}$, where $h_K(x)=max\{x\cdot y:y\in K\}$, for $x\in {\mathbb{R}}^n$. For the support function of the dilate $cK=\{cx:x\in K\}$ of a convex body K we have

$$h_{cK}=c{h}_K,c>0.$$

(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

$$K\subseteq L⟺h_K\le h_L.$$

(9)

For a convex body K and each Borel set $\omega \subset S^{n-1}$, the reverse spherical image $\tau (K,\omega )$, 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 $K\in {\mathcal{K}}_o^n$ there is a Borel measure $S_K$ on $S^{n-1}$ called the Aleksandrov-Fenchel surface area measure of K, defined by

$$S_K(\omega )={\mathcal{H}}^{n-1}(\tau (K,\omega )),$$

for each Borel set $\omega \subseteq {\mathbb{S}}^{n-1}$. Observe that for the surface area measure of the dilate cK of K we have

$$S_{cK}=c^{n-1}{S}_K,c>0.$$

The Minkowski sum of convex sets $K_1,\dots ,K_m$ in ${\mathbb{R}}^n$ is defined by

$$K_1+\cdots +K_m=\{x_1+\cdots +x_m:x_1\in K_1,\dots ,x_m\in K_m\}.$$

The mixed volume $V(K_1,\dots ,K_n)$ of compact convex sets $K_1,\dots ,K_n$ in ${\mathbb{R}}^n$ is defined by

$$V(K_1,\dots ,K_n)=\frac{1}{n!}\sum _{j=1}^n{(-1)}^{n+j}\sum _{i_1<\cdots <i_k}V(K_{i_1}+\cdots +K_{i_k}).$$

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

$$V_i{(K_1,K_2)}^2\ge V_{i+1}(K_1,K_2)V_{i-1}(K_1,K_2),$$

(10)

where

$$V_i(K_1,K_2)=V(\underset{n-i}{\underbrace{K_1,\dots ,K_1}},\underset{i}{\underbrace{K_2,\dots ,K_2}})$$

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 $K_1,\dots ,K_n\in \mathcal{K}$ and $1\le m\le n$. Then

$$V{(K_1,\dots ,K_n)}^m\ge \prod _{i=1}^{m}V(K_i,\dots ,K_i,K_{m+1},\dots ,K_n).$$

Hence

$$V_1{(K_1,K_2)}^{n-1}\ge V{(K_1)}^{n-2}{V}_{n-1}(K_1,K_2).$$

(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 $K_1\subset K_1^{\prime }$, then

$$V(K_1,K_2,\dots ,K_n)\le V(K_1^{\prime },K_2,\dots ,K_n).$$

The mixed volume $V_1(K,L)$ of the convex bodies $K,L\in {\mathcal{K}}_o^n$ 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

$$M(K)=\frac{2}{{\omega }_n}{V}_1(B,K),$$

that is,

$$M(K)=\frac{2}{n{\omega }_n}{\int }_{S^{n-1}}{h}_{K}d{S}_K.$$

The in-radius $r(K,L)$, out-radius $R(K,L)$ of K with respect to L are, respectively, defined by

$$\begin{array}{c}r(K,L)=sup\{\lambda :x\in {\mathbb{R}}^n\text{ and }x+\lambda L\subset K\},\hfill \\ R(K,L)=inf\{\lambda :x\in {\mathbb{R}}^n\text{ and }K\subset x+\lambda L\}.\hfill \end{array}$$

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 ${\mathbb{R}}^n$, then

$$V_1{(K,L)}^{n/(n-1)}-V(K)V{(L)}^{1/(n-1)}\ge {(V_1{(K,L)}^{1/(n-1)}-r(K,L)V{(L)}^{1/(n-1)})}^n,$$

(14)

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

Note that the right-hand side of (14) is non-negative for $x+r(K,L)L\subseteq K$ ($x\in {\mathbb{R}}^n$). By (13) we have

$$V_1(K,L)=\frac{1}{n}{\int }_{S^{n-1}}{h}_{L}d{S}_K\ge \frac{1}{n}{\int }_{S^{n-1}}{h}_{L}d{S}_{r(K,L)L}\ge r{(K,L)}^{n-1}V(L).$$

From Lemma 1 and using the inequality $x^{n-1}-y^{n-1}\ge {(x-y)}^{n-1}$ (for $x\ge y\ge 0$), a lower bound of the Minkowski deficit follows (cf. [26, 27]).

Proposition 1 Let K, L be convex bodies in the Euclidean space ${\mathbb{R}}^n$, then

$${\mathrm{\Delta }}_n(K,L)\ge {(V_1{(K,L)}^{1/(n-1)}-r(K,L)V{(L)}^{1/(n-1)})}^{n(n-1)},$$

(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 ${\mathbb{R}}^3$, then

$$\begin{array}{rcl}{\mathrm{\Delta }}_3(K,L)& \ge & {(V_1{(K,L)}^{1/2}-r(K,L)V{(L)}^{1/2})}^6\\ +2r{(K,L)}^{3}V{(L)}^{3/2}{(V_1{(K,L)}^{1/2}-r(K,L)V{(L)}^{1/2})}^3,\end{array}$$

(16)

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

Proof Since $V(K)\ge r{(K,L)}^{3}V(L)$ and by $x^3-y^3\ge {(x-y)}^3$ (for $x\ge y\ge 0$), we have

$$\begin{array}{rcl}{V}_1{(K,L)}^{3/2}+V(K)V{(L)}^{1/2}& \ge & V_1{(K,L)}^{3/2}+r{(K,L)}^{3}V{(L)}^{3/2}\\ =& V_1{(K,L)}^{3/2}-{(r(K,L)V{(L)}^{1/2})}^3+2r{(K,L)}^{3}V{(L)}^{3/2}\\ \ge & {(V_1{(K,L)}^{1/2}-r(K,L)V{(L)}^{1/2})}^3+2r{(K,L)}^{3}V{(L)}^{3/2}.\end{array}$$

Note that (14) can be rewritten as

$$V_1{(K,L)}^{3/2}-V(K)V{(L)}^{1/2}\ge {(V_1{(K,L)}^{1/2}-r(K,L)V{(L)}^{1/2})}^3.$$

Multiplying by $V_1{(K,L)}^{3/2}+V(K)V{(L)}^{1/2}$ on both sides, we have

$$V_1{(K,L)}^3-V{(K)}^{2}V(L)\ge {(V_1{(K,L)}^{1/2}-r(K,L)V{(L)}^{1/2})}^3(V_1{(K,L)}^{3/2}+V(K)V{(L)}^{1/2}).$$

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

Let L be the unit ball and notice $S(K)=3V_1(K,B)$ 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 ${\mathbb{R}}^3$ and r be the in-radius of K, then

$${\mathrm{\Delta }}_3(K)\ge {(S{(K)}^{1/2}-{(4\pi )}^{1/2}r)}^6+16{\pi }^{3/2}{r}^3{(S{(K)}^{1/2}-{(4\pi )}^{1/2}r)}^3,$$

(17)

with equality if and only if K is a ball.

For $n\ge 4$, we obtain a stronger Bonnesen-style Minkowski inequality as follows.

Theorem 2 Let K, L be convex bodies in the Euclidean space ${\mathbb{R}}^n$ ($n\ge 4$), then

$$\begin{array}{rcl}{\mathrm{\Delta }}_n(K,L)& \ge & {(V_1{(K,L)}^{1/(n-1)}-r(K,L)V{(L)}^{1/(n-1)})}^{n(n-1)}\\ +2{(r(K,L)V{(L)}^{1/(n-1)})}^{n(n-2)}{(V_1{(K,L)}^{1/(n-1)}-r(K,L)V{(L)}^{1/(n-1)})}^n\\ +{(V_1(K,L)V(L))}^{n/(n-1)}r{(K,L)}^n{(V_1{(K,L)}^{1/(n-1)}-r(K,L)V{(L)}^{1/(n-1)})}^{n(n-3)},\end{array}$$

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 $p\ge q$.

$$\begin{array}{c}\sum _{i=2}^n\left(V_1{(K,L)}^{n(n-i)/(n-1)}{(V{(K)}^{n-1}V(L))}^{(i-2)/(n-1)}\right)\hfill \\ \ge \sum _{i=2}^n\left(V_1{(K,L)}^{n(n-i)/(n-1)}{(V{(L)}^{1/(n-1)}r(K,L))}^{n(i-2)}\right)\hfill \\ =(p^{n-2}-q^{n-2})+pq(p^{n-4}+p^{n-5}q+\cdots +p^{n-i}{q}^{i-4}+\cdots +q^{n-4})+2q^{n-2}\hfill \\ =(p^{n-2}-q^{n-2})+pq\cdot \frac{p^{n-3}-q^{n-3}}{p-q}+2q^{n-2}\hfill \\ \ge {(p-q)}^{n-2}+pq{(p-q)}^{n-4}+2q^{n-2}\hfill \\ \ge {(p^{1/n}-q^{1/n})}^{n(n-2)}+pq{(p^{1/n}-q^{1/n})}^{n(n-4)}+2q^{n-2}.\hfill \end{array}$$

That is,

$$\begin{array}{c}\sum _{i=2}^n\left(V_1{(K,L)}^{n(n-i)/(n-1)}{(V{(K)}^{n-1}V(L))}^{(i-2)/(n-1)}\right)\hfill \\ \ge {(V_1{(K,L)}^{1/(n-1)}-V{(L)}^{1/(n-1)}r(K,L))}^{n(n-2)}+2{(V{(L)}^{1/(n-1)}r(K,L))}^{n(n-2)}\hfill \\ +{(V_1(K,L)V(L))}^{n/(n-1)}r{(K,L)}^n{(V_1{(K,L)}^{1/(n-1)}-V{(L)}^{1/(n-1)}r(K,L))}^{n(n-4)}.\hfill \end{array}$$

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

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

we obtain

$$\begin{array}{rcl}{V}_1{(K,L)}^n-V{(K)}^{n-1}V(L)& \ge & {(V_1{(K,L)}^{1/(n-1)}-r(K,L)V{(L)}^{1/(n-1)})}^n\\ \times \left(\sum _{i=2}^n\left(V_1{(K,L)}^{n(n-i)/(n-1)}{(V{(K)}^{n-1}V(L))}^{(i-2)/(n-1)}\right)\right).\end{array}$$

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 $n\ge 4$.

Corollary 2 Let K be a convex body in ${\mathbb{R}}^n$ ($n\ge 4$) and r be the in-radius of K, then

$$\begin{array}{rcl}{\mathrm{\Delta }}_n(K)& \ge & {(S{(K)}^{1/(n-1)}-{(n{\omega }_n)}^{1/(n-1)}r)}^{n(n-1)}\\ +2{\left({(n{\omega }_n)}^{1/(n-1)}r\right)}^{n(n-2)}{(S{(K)}^{1/(n-1)}-{(n{\omega }_n)}^{1/(n-1)}r)}^n\\ +{(n{\omega }_{n}S(K))}^{n/(n-1)}{r}^n{(S{(K)}^{1/(n-1)}-{(n{\omega }_n)}^{1/(n-1)}r)}^{n(n-3)},\end{array}$$

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 ${\mathbb{R}}^n$, then

$$\frac{V_0(K,L)}{V_1(K,L)}\le {\left(\frac{V_0(K,L)}{V_n(K,L)}\right)}^{1/n}\le \frac{V_{n-1}(K,L)}{V_n(K,L)}\le \frac{V_1{(K,L)}^{n-1}}{V{(K)}^{n-2}V(L)},$$

(18)

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

Proof By inequality (12), we have

$$\frac{V_{n-1}(K,L)}{V(L)}\le \frac{V_1{(K,L)}^{n-1}}{V{(K)}^{n-2}V(L)}.$$

By the Aleksandrov-Fenchel inequality (10) we have

$$\frac{V_0(K,L)}{V_1(K,L)}\le \frac{V_1(K,L)}{V_2(K,L)}\le \cdots \le \frac{V_i(K,L)}{V_{i+1}(K,L)}\le \cdots \le \frac{V_{n-1}(K,L)}{V_n(K,L)}.$$

Therefore

$$\frac{V_0(K,L)}{V_1(K,L)}\le {\left(\frac{V_0(K,L)}{V_n(K,L)}\right)}^{1/n}\le \frac{V_{n-1}(K,L)}{V_n(K,L)}.$$

 □

Theorem 3 Let K, L be convex bodies in ${\mathbb{R}}^n$, then

$${\mathrm{\Delta }}_n(K,L)\ge V{(K)}^{n-2}V(L)V_1(K,L)(\frac{V_{n-1}(K,L)}{V_n(K,L)}-{\left(\frac{V_0(K,L)}{V_n(K,L)}\right)}^{1/n}),$$

(19)

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

Proof Via (18), we have

$$\frac{V_1{(K,L)}^{n-1}}{V{(K)}^{n-2}V(L)}-\frac{V_0(K,L)}{V_1(K,L)}\ge \frac{V_{n-1}(K,L)}{V_n(K,L)}-{\left(\frac{V_0(K,L)}{V_n(K,L)}\right)}^{1/n}.$$

That is

$$V_1{(K,L)}^n-V{(K)}^{n-1}V(L)\ge V{(K)}^{n-2}V(L)V_1(K,L)(\frac{V_{n-1}(K,L)}{V_n(K,L)}-{\left(\frac{V_0(K,L)}{V_n(K,L)}\right)}^{1/n}).$$

 □

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

Theorem 4 Let K be a convex body in ${\mathbb{R}}^n$, then

$${\mathrm{\Delta }}_n(K)\ge n^{n-1}{\omega }_{n}S(K)V{(K)}^{n-2}(\frac{M(K)}{2}-{\left(\frac{V(K)}{{\omega }_n}\right)}^{1/n}),$$

with equality if and only if K is a ball.

Lemma 3 Let K, L be convex bodies in ${\mathbb{R}}^n$, then

$$\begin{array}{c}\frac{V_1{(K,L)}^{n-1}-\sqrt{V_1{(K,L)}^{n-2}(V_1{(K,L)}^n-V{(K)}^{n-1}V(L))}}{V{(K)}^{n-2}V(L)}\hfill \\ \le \frac{V_0(K,L)}{V_1(K,L)}\le \frac{V_{n-1}(K,L)}{V_n(K,L)}\le \frac{V_1{(K,L)}^{n-1}}{V{(K)}^{n-2}V(L)}.\hfill \end{array}$$

(20)

Proof The Minkowski inequality (5) gives

$$\frac{V_1{(K,L)}^{n-1}-\sqrt{V_1{(K,L)}^{n-2}(V_1{(K,L)}^n-V{(K)}^{n-1}V(L))}}{V{(K)}^{n-2}V(L)}\le \frac{V_0(K,L)}{V_1(K,L)}.$$

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 ${\mathbb{R}}^n$, then

$${\mathrm{\Delta }}_n(K,L)\ge \frac{V{(K)}^{2n-4}V{(L)}^2}{V_1{(K,L)}^{n-2}}{(\frac{V_{n-1}(K,L)}{V_n(K,L)}-\frac{V_0(K,L)}{V_1(K,L)})}^2,$$

(21)

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

Proof From (20) we have

$$\frac{\sqrt{V_1{(K,L)}^{n-2}(V_1{(K,L)}^n-V{(K)}^{n-1}V(L))}}{V{(K)}^{n-2}V(L)}\ge \frac{V_{n-1}(K,L)}{V_n(K,L)}-\frac{V_0(K,L)}{V_1(K,L)}.$$

 □

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 ${\mathbb{R}}^n$, then

$$S{(K)}^n-n^n{\omega }_{n}V{(K)}^{n-1}\ge \frac{n^{2n-2}{\omega }_n^2{V}^{2n-4}}{S^{n-2}}{(\frac{M(K)}{2}-\frac{nV(K)}{S(K)})}^2,$$

with equality if and only if K is a ball.