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Some inequalities related to (i,j)type {L}_{p}mixed affine surface area and {L}_{p}mixed curvature image
Journal of Inequalities and Applications volume 2013, Article number: 470 (2013)
Abstract
In this article, we introduce two concepts: the (i,0)type {L}_{p}mixed affine surface area and (i,j)type {L}_{p}mixed affine surface area in the set of convex bodies such that {L}_{p}affine surface area by Lutwak et al. is proposed in its special cases. Besides, applying these concepts, we establish the extension results of the wellknown {L}_{p}Petty affine projection inequality, {L}_{p}Busemann centroid inequality and its dual inequality.
MSC:52A40, 52A20.
1 Introduction
Let {\mathcal{K}}^{n} denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space {\mathbb{R}}^{n}. For the set of convex bodies, which contain the origin in their interiors, and the set of originsymmetric convex bodies in {\mathcal{K}}^{n}, we write {\mathcal{K}}_{o}^{n} and {\mathcal{K}}_{e}^{n}, respectively. Let {S}_{o}^{n} denote the set of star bodies (about the origin) in {\mathbb{R}}^{n}, and let {S}_{e}^{n} denote the set of originsymmetric star bodies in {\mathcal{S}}_{o}^{n}. Let {S}^{n1} denote the unit sphere in {\mathbb{R}}^{n}, and let V(K) denote the ndimensional volume of body K. If K is the standard unit ball B in {\mathbb{R}}^{n}, then it is denoted by {\omega}_{n}=V(B). Note that {\omega}_{n}=\frac{{\pi}^{n/2}}{\mathrm{\Gamma}(1+n/2)} defines {\omega}_{n} for all nonnegative real n (not just the positive integers).
A body K\in {\mathcal{K}}^{n} is said to have a continuous i th curvature function {f}_{i}(K,\cdot ):{S}^{n1}\to [0,\mathrm{\infty}) if and only if {S}_{i}(K,\cdot ) is absolutely continuous with respect to S and has the RadonNikodym derivative (see [1])
Let {\mathcal{F}}_{i}^{n} denote the subset of all bodies {\mathcal{K}}^{n} which have a positive continuous i th curvature function. Let {\mathcal{F}}_{i,o}^{n}, {\mathcal{F}}_{i,e}^{n} denote the set of all bodies in {\mathcal{K}}_{o}^{n}, {\mathcal{K}}_{e}^{n}, respectively, and both of them have a positive continuous i th curvature function.
A convex body K\in {\mathcal{K}}_{o}^{n} is said to have an {L}_{p}curvature function {f}_{p}(K,\cdot ):{S}^{n1}\to \mathbb{R} if its {L}_{p}surface area measure {S}_{p}(K,\cdot ) is absolutely continuous with respect to the spherical Lebesgue measure S, and it has the RadonNikodym derivative (see [2])
For p\ge 1, K\in {\mathcal{F}}_{o}^{n}, then {L}_{p}affine surface area {\mathrm{\Omega}}_{p}(K) of K by (see [3–5])
For each K\in {\mathcal{K}}^{n} and p\ge 1, the {L}_{p}projection body, {\mathrm{\Pi}}_{p}K, of K is an originsymmetric convex body whose support function is given by (see [6])
where {c}_{n,p}=\frac{{\omega}_{n+p}}{{\omega}_{2}{\omega}_{n}{\omega}_{p1}}. When p=1, (1.4) is the notion of projection body (see [7]).
It is easy to show that if E is an ellipsoid which is centered at the origin, then (see [[8], p.105])
The wellknown {L}_{p}Petty affine projection inequality is expressed as follows (see [8–10]).
Theorem A ({L}_{p}Petty affine projection inequality)
If K,L\in {\mathcal{F}}_{o}^{n}, p\ge 1, then
with equality if and only if K is an ellipsoid which is centered at the origin.
Let K\in {\mathcal{S}}_{o}^{n}, and let p\ge 1, then the {L}_{p}centroid body, {\mathrm{\Gamma}}_{p}K, of K is the originsymmetric convex body whose support function is given by (see [6, 11])
If E is an ellipsoid which is centered at the origin, then {\mathrm{\Gamma}}_{p}E=E. In particular, {\mathrm{\Gamma}}_{p}B=B.
The wellknown {L}_{p}BusemannPetty centroid inequality is as follows (see [6]).
Theorem B ({L}_{p}BusemannPetty centroid inequality)
If K\in {S}_{o}^{n} and p\ge 1, then
with the equality if and only if K is an ellipsoid which is centered at the origin.
Lutwak et al. introduced the concept of dual {L}_{p}centroid bodies (see [12]). We give the concept of the unusual normalization of dual {L}_{p}centroid bodies such that {\mathrm{\Gamma}}_{p}B=B: Let K\in {\mathcal{K}}_{o}^{n} and real p>0, then radial function of dual {L}_{p}centroid body, {\mathrm{\Gamma}}_{p}K, of K is defined by
It is easy to show that if E is an ellipsoid which is centered at the origin, then
Combined with (1.5) and (1.10), we have that {\mathrm{\Gamma}}_{p}E=E. In particular, {\mathrm{\Gamma}}_{p}B=B.
Si Lin gives the following dual inequality of inequality (1.8) (see [[13], p.9, Theorem 5.4]).
Theorem C (Dual {L}_{p}BusemannPetty centroid inequality)
If K\in {\mathcal{K}}_{o}^{n} and p\ge 1, then
with equality if and only if K is an ellipsoid which is centered at the origin.
Liu et al. [14], Lu and Wang [15], Ma and Liu [16, 17] independently proposed the notion of {L}_{p}mixed curvature function: Let p\ge 1, i=0,1,\dots ,n1, a convex body K\in {\mathcal{K}}_{o}^{n} is said to have an {L}_{p}mixed curvature function {f}_{p,i}(K,\cdot ):{S}^{n1}\to \mathbb{R}, if its {L}_{p}mixed surface area measure {S}_{p,i}(K,\cdot ) is absolutely continuous with respect to spherical Lebesgue measure S and has the RadonNikodym derivative
If the mixed surface area measure {S}_{i}(K,\cdot ) is absolutely continuous with respect to spherical Lebesgue measure S, we have
According to the concept of {L}_{p}mixed curvature function of convex body, Lu and Wang [15] and Ma introduce the concept of {L}_{p}mixed curvature image of convex body as follows: For each K\in {\mathcal{F}}_{i,o}^{n} (i=0,1,\dots ,n1) and real p\ge 1, define {\mathrm{\Lambda}}_{p,i}K\in {\mathcal{S}}_{o}^{n}, the {L}_{p}mixed curvature image of K, by
If i=0 in (1.14), then {\mathrm{\Lambda}}_{p,0}K={\mathrm{\Lambda}}_{p}K. The unusual normalization of definition (1.14) is chosen so that for the unit ball B, we have {\mathrm{\Lambda}}_{p,i}B=B. For K\in {\mathcal{F}}_{i,o}^{n}, ni\ne p\ge 1, \lambda >0,
Let {\mathcal{C}}_{i,o}^{n} denote the set of {L}_{p}mixed curvature images of convex bodies in {\mathcal{F}}_{i,o}^{n}. That is,
Because the {L}_{p}mixed curvature image belongs to star bodies, thus, {\mathcal{C}}_{i,o}^{n}\subseteq {\mathcal{S}}_{o}^{n}.
For each K\in {\mathcal{K}}^{n}, real p\ge 1 and i=0,1,\dots ,n1, the {L}_{p}mixed projection body, {\mathrm{\Pi}}_{p,i}K, of K is an originsymmetric convex body whose support function is given by (see [18])
where {S}_{p,i}(K,\cdot ) (i=0,1,\dots ,n1) is {L}_{p}mixed surface area measure. Obviously, {\mathrm{\Pi}}_{p,0}K={\mathrm{\Pi}}_{p}K, and for the standard unit ball B, we have {\mathrm{\Pi}}_{p,i}B=B. For K\in {\mathcal{K}}^{n}, \lambda >0, p\ge 1 and 0\le i<n, then
Let K\in {\mathcal{S}}_{o}^{n}, real p\ge 1 and i be arbitrary real numbers, then the {L}_{p}mixed centroid body, {\mathrm{\Gamma}}_{p,i}K, of K is the originsymmetric convex body whose support function is given by (see [19])
Obviously, {\mathrm{\Gamma}}_{p,0}K={\mathrm{\Gamma}}_{p}K, and for the standard unit ball B, we have {\mathrm{\Gamma}}_{p,i}B=B.
Ma introduced the concept of dual {L}_{p}mixed centroid body (see [19]). Further, we introduced the concept of the unusual normalization of dual {L}_{p}mixed centroid body {\mathrm{\Gamma}}_{p,i}K as follows: Let K\in {\mathcal{K}}_{o}^{n}, p>0, i=0,1,\dots ,n1, then the dual {L}_{p}mixed centroid bodies, {\mathrm{\Gamma}}_{p,i}K, of K are defined by:
Obviously, {\mathrm{\Gamma}}_{p,0}K={\mathrm{\Gamma}}_{p}K, and for the standard unit ball B, we have {\mathrm{\Gamma}}_{p,i}B=B.
In this article, we will first introduce the concept of (i,0)type {L}_{p}mixed affine surface area of convex body as follows.
Definition 1.1 For K\in {\mathcal{F}}_{i,o}^{n}, i=0,1,\dots ,n1 and p\ge 1, the (i,0)type {L}_{p}mixed affine surface area, {\mathrm{\Omega}}_{p}^{(i)}(K), of K is defined by
Next, we have established an extension of {L}_{p}Petty affine projection inequality (1.6) as follows.
Theorem 1.1 Let K\in {\mathcal{F}}_{i,o}^{n}, i=0,1,\dots ,n1 and p\ge 1, then
with equality in inequality (1.21) for 0<i<n1 if and only if K is a ball which is centered at the origin; for i=0 if and only if K is an ellipsoid which is centered at the origin.
Further, we obtain the following generalized {L}_{p}BusemannPetty centroid inequality.
Theorem 1.2 Suppose that K\in {C}_{i,o}^{n}\subseteq {\mathcal{S}}_{o}^{n}, i=0,1,\dots ,n1 and p\ge 1, then
with equality in inequality (1.22) for 0<i<n1 if and only if K is a ball which is centered at the origin; for i=0 if and only if K is an ellipsoid which is centered at the origin.
Finally, we get the following dual inequality of the inequality (1.22).
Theorem 1.3 Suppose that K\in {\mathcal{K}}_{o}^{n}. If {\mathrm{\Gamma}}_{p,i}K\in {C}_{i,o}^{n}\subseteq {\mathcal{S}}_{o}^{n}, i=0,1,\dots ,n1 and p\ge 1, then
with equality in inequality (1.23) for 0<i<n1 if and only if K is a ball which is centered at the origin; for i=0 if and only if K is an ellipsoid which is centered at the origin.
2 Preliminaries
2.1 Support function, radial function and polar of convex body
If K\in {\mathcal{K}}^{n}, then its support function, {h}_{K}=h(K,\cdot ):{\mathbb{R}}^{n}\to (0,\mathrm{\infty}), is defined by (see [20, 21])
Obviously, if K\in {\mathcal{K}}^{n}, λ is a positive constant, x\in {\mathbb{R}}^{n}, then h(\lambda K,x)=\lambda h(K,x).
If K is a compact starshaped (about the origin) in {\mathbb{R}}^{n}, its radial function, {\rho}_{K}=\rho (K,\cdot ):{\mathbb{R}}^{n}\setminus \{0\}\to [0,\mathrm{\infty}), is defined by (see [20, 21])
When {\rho}_{K} is positive and continuous, K is called a star body (about the origin). Obviously, if K\in {\mathcal{S}}_{o}^{n}, \alpha >0, x\in {\mathbb{R}}^{n}, then \rho (K,\alpha x)={\alpha}^{1}\rho (K,x) and \rho (\alpha K,x)=\alpha \rho (K,x). Two star bodies K and L are said to be dilates (of one another) if {\rho}_{K}(u)/{\rho}_{L}(u) is independent on u\in {S}^{n1}.
For K\in {\mathcal{K}}_{o}^{n}, the polar body, {K}^{\ast}, of K is defined by (see [20, 21])
Obviously, we have {({K}^{\ast})}^{\ast}=K. If \lambda >0, then
If K\in {\mathcal{K}}_{o}^{n}, then the support and radial function of the polar body {K}^{\ast}, of K are defined respectively by (see [20, 21])
for all u\in {S}^{n1}.
2.2 The quermassintegrals, {L}_{p}mixed quermassintegrals and {L}_{p}mixed volume
For K\in {\mathcal{K}}^{n} and i=0,1,\dots ,n1, the quermassintegrals, {W}_{i}(K), of K are defined by (see [20, 21])
From (2.3), we easily see that
For p\ge 1, K,L\in {\mathcal{K}}_{o}^{n} and \epsilon >0, the Firey {L}_{p}combination K{+}_{p}\epsilon \cdot L\in {\mathcal{K}}_{o}^{n} is defined by (see [22])
where ‘⋅’ in \epsilon \cdot L denotes the Firey scalar multiplication.
For K,L\in {\mathcal{K}}_{o}^{n}, \epsilon >0 and real p\ge 1, the {L}_{p}mixed quermassintegrals, {W}_{p,i}(K,L), of K and L (i=0,1,\dots ,n1) are defined by (see [1])
Obviously, for p=1, {W}_{1,i}(K,L) is just the classical mixed quermassintegral {W}_{i}(K,L). For i=0, the {L}_{p}mixed quermassintegral {W}_{p,0}(K,L) is just the {L}_{p}mixed volume {V}_{p}(K,L), namely,
For p\ge 1, i=0,1,\dots ,n1 and each K\in {\mathcal{K}}_{o}^{n}, there exists a positive Borel measure {S}_{p,i}(K,\cdot ) on {S}^{n1} such that the {L}_{p}mixed quermassintegral {W}_{p,i}(K,L) has the following integral representation (see [1]):
for all L\in {\mathcal{K}}_{o}^{n}. It turns out that the measure {S}_{p,i}(K,\cdot ) (i=0,1,\dots ,n1) on {S}^{n1} is absolutely continuous with respect to {S}_{i}(K,\cdot ) and has the RadonNikodym derivative
From (2.3) and (2.7), it follows immediately that for each K\in {\mathcal{K}}_{o}^{n},
If K,L\in {\mathcal{K}}_{o}^{n}, p\ge 1, by definition (1.12), then formula (2.7) of the {L}_{p}mixed quermassintegral can be rewritten as follows:
We shall require the Minkowski inequality for the {L}_{p}mixed quermassintegrals {W}_{p,i} as follows (see [1]): For K,L\in {\mathcal{K}}_{o}^{n}, and p\ge 1, 0\le i<n, then
with equality for p=1 and 0\le i<n1 if and only if K and L are homothetic; for p>1 and 0\le i\le n1 if and only if K and L are dilates. For p=1 and i=n1, inequality (2.11) is identical.
2.3 Dual quermassintegrals and {L}_{p}dual mixed quermassintegrals
For K\in {S}_{o}^{n} and any real i, the dual quermassintegrals, {\tilde{W}}_{i}(K), of K are defined by (see [20, 21])
Obviously,
For K,L\in {S}_{o}^{n}, p\ge 1 and \epsilon >0, the {L}_{p}harmonic radial combination K{+}_{p}\epsilon \cdot L\in {S}_{o}^{n} is defined by (see [2, 23, 24])
Note that here ‘\epsilon \cdot L’ is different from ‘\epsilon \cdot L’ in the Firey {L}_{p}combination.
For K,L\in {S}_{o}^{n}, \epsilon >0, p\ge 1 and real i\ne n, the {L}_{p}dual mixed quermassintegrals, {\tilde{W}}_{p,i}(K,L), of K and L are defined by [25]
If i=0, we easily see that definition (2.14) is just the definition of {L}_{p}dual mixed volume, namely,
From definition (2.14), the integral representation of the {L}_{p}dual mixed quermassintegrals is given by (see [25]): If K,L\in {S}_{o}^{n}, p\ge 1, and real i\ne n, i\ne n+p, then
Together with (2.12) and (2.16), for K\in {S}_{o}^{n}, p\ge 1, and i\ne n,n+p, we get
Analog of the Minkowski inequality for {L}_{p}dual mixed quermassintegrals is as follows (see [25]): If K,L\in {S}_{o}^{n}, p\ge 1, then for i<n or i>n+p,
For n<i<n+p, inequality (2.18) is reversed. With equality in every inequality if and only if K and L are dilates.
3 The (i,j)type {L}_{p}mixed affine surface area
In this section, we further propose the concept of (i,j)type {L}_{p}mixed affine surface area as follows.
Definition 3.1 For K\in {\mathcal{F}}_{i,o}^{n}, i=0,1,\dots ,n1, j\in \mathbb{R} and p\ge 1, the (i,j)type {L}_{p}mixed affine surface area, {\mathrm{\Omega}}_{p,j}^{(i)}(K), of K is defined by
Obviously, {\mathrm{\Omega}}_{p}^{(0)}(K)={\mathrm{\Omega}}_{p}(K) and {\mathrm{\Omega}}_{p,0}^{(i)}(K)={\mathrm{\Omega}}_{p}^{(i)}(K).
Next, we introduce the concept of (i,0)type {L}_{p}mixed affine surface area of the convex bodies {K}_{1},{K}_{2},\dots ,{K}_{ni} as follows.
Definition 3.2 For p\ge 1, i=0,1,\dots ,n1, the (i,0)type {L}_{p}mixed affine surface area, {\mathrm{\Omega}}_{p}^{(i)}({K}_{1},\dots ,{K}_{ni}), of {K}_{1},\dots ,{K}_{ni}\in {\mathcal{F}}_{i,o}^{n} is defined by
Let {K}_{1}=\cdots ={K}_{nij}=K and {K}_{nij+1}=\cdots ={K}_{ni}=L (j=0,\dots ,ni), we define {\mathrm{\Omega}}_{p,j}^{(i)}(K,L)={\mathrm{\Omega}}_{p}^{(i)}(K,\dots ,K,L,\dots ,L) with nij copies of K and j copies of L. From this, if j is any real number, we can define the following.
Definition 3.3 For K,L\in {\mathcal{F}}_{i,o}^{n}, i=0,\dots ,n1, p\ge 1, j\in \mathbb{R}, the (i,j)type {L}_{p}mixed affine surface area, {\mathrm{\Omega}}_{p,j}^{(i)}(K,L), of K, L is defined by
Specially for the case j=p, we have that
Take L=B in (3.4) and write
Because for u\in {S}^{n1}, {S}_{i}(B,u)=S, h(B,u)=1, so by (2.8) and (1.13), we get {f}_{p,i}(B,u)=1. This together with (3.3) and (3.5) yields
and {\mathrm{\Omega}}_{p,j}^{(i)}(K) is called the (i,j)type {L}_{p}mixed affine surface area of K\in {\mathcal{F}}_{i,o}^{n}. In particular, {\mathrm{\Omega}}_{p,j}^{(o)}(K)={\mathrm{\Omega}}_{p,j}(K) is called the j th {L}_{p}mixed affine surface area of K\in {\mathcal{F}}_{o}^{n} (see [26]).
Next, we give some propositions of {L}_{p}mixed curvature image and (i,j)type {L}_{p}mixed affine surface area.
Proposition 3.1 Let K\in {\mathcal{F}}_{i,o}^{n}, i=0,1,\dots ,n1, j\in \mathbb{R}. Then
In particular, take j=0 in (3.7), then
Proof From (3.6), (1.14) and (2.12), we have
□
Proposition 3.2 Let p\ge 1, K\in {\mathcal{F}}_{i,o}^{n} and i=0,1,\dots ,n1. Then
for each Q\in {\mathcal{K}}_{o}^{n}.
Proof For each Q\in {\mathcal{K}}_{o}^{n}, from (2.10), (1.14), (2.2) and (2.16), we have
□
Proposition 3.3 If p\ge 1, L\in {\mathcal{F}}_{i,o}^{n}, then
for all K\in {\mathcal{S}}_{o}^{n} with equality if and only if K and {\mathrm{\Lambda}}_{p,i}L are dilates.
Proof Let L\in {\mathcal{F}}_{i,o}^{n} and each K\in {\mathcal{S}}_{o}^{n}, then from (1.20), (2.2), (2.7), (2.12) and Hölder’s inequality, we have
From this, we immediately get (3.10).
According to the condition of equality to hold in the Hölder inequality, we know that equality holds in (3.10) if and only if
for any u\in {S}^{n1}, where c is a constant. Combined with the definition of {L}_{p}mixed curvature image, for any u\in {S}^{n1}, we have
this shows that K and {\mathrm{\Lambda}}_{p,i}L are dilates. Therefore, the equality holds in inequality (3.10) if and only if K and {\mathrm{\Lambda}}_{p,i}L are dilates. The proof is complete. □
Now, according to Proposition 3.3, we can give an expansion of the definition of the (i,0)type {L}_{p}mixed affine surface area of K\in {\mathcal{K}}_{o}^{n} as follows.
Definition 3.4 If K\in {\mathcal{K}}_{o}^{n}, p\ge 1, then the (i,0)type {L}_{p}mixed affine surface area, {\mathrm{\Omega}}_{p}^{(i)}(K), of K is defined by
For i=0, the definition is just the definition of {L}_{p}affine surface area by Lutwak proposed in [2].
4 Generalized {L}_{p}Petty affine projection inequality
In this section, we complete the proof of Theorem 1.1 in the introduction. In fact, we prove the following more general conclusion.
Theorem 4.1 Let K\in {\mathcal{K}}_{o}^{n}, L\in {\mathcal{F}}_{i,o}^{n}, p\ge 1, 0\le i<n1, then
with equality in inequality (4.1) for 0<i<n1 if and only if K and L are balls of dilates which are centered at the origin; for i=0 if and only if K and L are ellipsoids of dilates which are centered at the origin.
In order to prove the theorems above, we first give the following three lemmas.
Lemma 4.1 (See [27])
Suppose that K\in {\mathcal{K}}_{o}^{n}, i\in \mathbb{R} and 0\le i<n, then
with the equality for 0<i<n if and only if K is a ball which is centered at the origin. If i=0, then (4.2) is identical.
Lemma 4.2 (See [18])
Suppose that K\in {\mathcal{K}}_{o}^{n}, p>1 and 0<i<n1, i is a positive integer, then
with equality if and only if K is a ball which is centered at the origin.
Remark 4.1 The conditions of inequality (4.3) can be relaxed to p\ge 1 and 0\le i<n1, while the conditions of the equality that holds can be given separately. For 0<i<n1 and p=1, the inequality (4.3) is proved by Lutwak with the equality holding if and only if K is a ball (see [7]). For i=0 and p>1, inequality (4.3) is proved by Lutwak et al. with the equality that holds if and only if K is an ellipsoid which is centered at the origin (see [6]). For i=0 and p=1, then (4.3) is the famous Petty projection inequality (see [28]), with the equality that holds if and only if K is an ellipsoid.
Lemma 4.3 If K,L\in {\mathcal{K}}_{o}^{n}, p\ge 1, i=0,1,\dots ,n1, then
Proof From (1.16), (2.10) and the Fubini theorem, it is easy to prove Lemma 4.3. □
Proof of Theorem 4.1 For L\in {\mathcal{F}}_{i,o}^{n} and any Q\in {\mathcal{K}}_{o}^{n}, by inequality (3.10) and Lemma 4.1, we have
with equality for 0<i\le n1 if and only if {\mathrm{\Lambda}}_{p,i}L and {Q}^{\ast} are centered balls of dilates; for i=0 if and only if {\mathrm{\Lambda}}_{p,i}L and {Q}^{\ast} are dilates.
Take Q={\mathrm{\Pi}}_{p,i}K with K\in {\mathcal{K}}_{o}^{n} in (4.5), then
with equality for 0<i\le n1 if and only if {\mathrm{\Lambda}}_{p,i}L and {\mathrm{\Pi}}_{p,i}^{\ast}K are centered balls of dilates; for i=0 if and only if {\mathrm{\Lambda}}_{p}L and {\mathrm{\Pi}}_{p}^{\ast}K are dilates.
Combining with inequalities (4.3) and (4.6), we give
which implies that inequality (4.1) holds.
Next, we discuss the conditions of equality that holds in inequality (4.1).
According to the condition of the equality that holds in inequality (4.3) and inequality (4.6) with Remark 4.1, the four steps will be given.

(1)
For the case p>1 and 0<i<n1, the equality holds in (4.1) if and only if {\mathrm{\Pi}}_{p,i}^{\ast}K and {\mathrm{\Lambda}}_{p,i}L are balls of dilates which are centered at the origin, and K is a ball which is centered at the origin. Together with {\mathrm{\Lambda}}_{p,i}B=B and {\mathrm{\Pi}}_{p,i}B=B, we know that K and L are balls of dilates which are centered at the origin.

(2)
For the case p=1 and 0<i<n1, the equality holds in (4.1) if and only if {\mathrm{\Pi}}_{1,i}^{\ast}K and {\mathrm{\Lambda}}_{1,i}L are balls of dilates which are centered at the origin, and K is a ball. By using {\mathrm{\Pi}}_{p,i}B=B and (1.17), it is obtained that {\mathrm{\Pi}}_{1,i}^{\ast}(\lambda B)={\lambda}^{1+in}{\mathrm{\Pi}}_{1,p}^{\ast}B={\lambda}^{1+in}B (\lambda >0) is a ball which is centered at the origin. Because {\mathrm{\Pi}}_{1,i}^{\ast}K and {\mathrm{\Lambda}}_{1,i}L are balls of dilates which are centered at the origin, then {\mathrm{\Lambda}}_{1,i}L is a ball of dilates which are centered at the origin, and together with {\mathrm{\Lambda}}_{p,i}B=B and (1.15), L is a ball which is centered at the origin. However, K is a ball, so the equality holds in (4.1) if and only if K and L are balls of dilates which are centered at the origin.

(3)
For the case p>1 and i=0, the equality holds in (4.1) if and only if {\mathrm{\Pi}}_{p}^{\ast}K and {\mathrm{\Lambda}}_{p}L are dilates and K is an ellipsoid which is centered at the origin. Let K=E be an ellipsoid which is centered at the origin, from (1.5), we know that {\mathrm{\Pi}}_{p}^{\ast}E={({\omega}_{n}/V(E))}^{\frac{1}{p}}E is an ellipsoid which is centered at the origin. Other, from the literature [2], we know that L is an ellipsoid E which is centered at the origin if and only if {\mathrm{\Lambda}}_{p}L are dilates of polar body {E}^{\ast} of this E. So we know that the equality holds in (4.1) if and only if L and K are ellipsoids which are centered at the origin and both are dilates.

(4)
For the case p=1 and i=0, the equality holds in (4.1) if and only if {\mathrm{\Lambda}}_{1}L and {\mathrm{\Pi}}_{1}^{\ast}K are dilates, and K is an ellipsoid. Suppose that K=\lambda E+{x}_{0} with \lambda >0, {x}_{0}\in {\mathbb{R}}^{n}, and E is an ellipsoid which is centered at the origin, noting that S(\lambda E+{x}_{0},\cdot )=S(\lambda E,\cdot )={\lambda}^{n1}S(E,\cdot ) (see [29]), this together with (1.5) {\mathrm{\Pi}}_{1}^{\ast}K={\mathrm{\Pi}}_{1}^{\ast}(\lambda E+{x}_{0})={\mathrm{\Pi}}_{1}^{\ast}\lambda E={\lambda}^{1n}{\mathrm{\Pi}}_{1}^{\ast}E={\lambda}^{1n}({\omega}_{n}/V(E))E is an ellipsoid which is centered at the origin. Because {\mathrm{\Lambda}}_{1}L and {\mathrm{\Pi}}_{1}^{\ast}K are dilates, then {\mathrm{\Lambda}}_{1}L is an ellipsoid which is centered at the origin. However, from [2], we know that L is an ellipsoid E which is centered at the origin if and only if {\mathrm{\Lambda}}_{1}L are dilates of polar body {E}^{\ast} of this E. Therefore, the equality holds in (4.1) if and only if K and L are ellipsoids of the dilates which are centered at the origin.
To sum up, the equality holds in (4.1) for p\ge 1 and 0<i<n1 if and only if K and L are balls of the dilates which are centered at the origin; for p\ge 1 and i=0 if and only if K and L are ellipsoids of the dilates which are centered at the origin. The proof is complete. □
Proof of Theorem 1.1 Exchange K and L in inequality (4.1), we have for L\in {\mathcal{K}}_{o}^{n}, K\in {\mathcal{F}}_{i,o}^{n}, p\ge 1, 0\le i<n1
By using Lemmas 4.3 and (4.7), we have
Taking L={\mathrm{\Pi}}_{p,i}K in the inequality above, we immediately obtain inequality (1.21). The proof is complete. □
Combining with Theorem 1.1 and (3.8), we immediately obtain the following.
Corollary 4.1 If K\in {\mathcal{F}}_{i,o}^{n}, i=0,1,\dots ,n1 and p\ge 1, then
with the equality in inequality (4.8) for 0<i<n1 if and only if K is a ball which is centered at the origin; for i=0 if and only if K is an ellipsoid which is centered at the origin.
Further, we have established the following results.
Theorem 4.2 Let K,L\in {\mathcal{F}}_{i,o}^{n}, 0\le i<n1, p\ge 1, then
with the equality in inequality (4.9) for 0<i<n1 if and only if K and L are balls of dilates which are centered at the origin; for i=0 if and only if K and L are ellipsoids of dilates which are centered at the origin.
Proof From inequality (4.1), we know that for Q\in {\mathcal{K}}_{o}^{n}, L\in {\mathcal{F}}_{i,o}^{n}, p\ge 1, 0\le i\le n1,
Take Q={\mathrm{\Pi}}_{p,i}K in (4.9), and using Corollary 4.1 and Lemma 4.3, we have
Combining inequality (4.11) with (3.8), we immediately obtain inequality (4.8). According to the condition of the equality holding in inequalities (4.1) and (4.9), the condition of the equality that holds in inequality (4.8) is easily obtained. The proof is complete. □
5 Generalized {L}_{p}BusemannPetty centroid inequality and dual inequality
In this section, we give the extension of the wellknown {L}_{p}BusemannPetty centroid inequality (1.8). Namely, we complete the proof of Theorem 1.2 and Theorem 1.3 (i.e., dual inequality of Theorem 1.2) in the introduction.
Lemma 5.1 If K\in {\mathcal{F}}_{i,o}^{n}, 0=0,1,\dots ,n1, p\ge 1, then
Proof Using definition (1.16) of {L}_{p}mixed projection body and definition (1.14) of {L}_{p}mixed curvature image, it is easy to prove (5.1). □
Lemma 5.2 If K\in {\mathcal{K}}_{o}^{n}, L\in {\mathcal{S}}_{o}^{n}, p\ge 1, i=0,1,\dots ,n1, then
Proof By (1.18), (1.19), (2.3), (2.7), (2.12) and (2.16), it is easy to prove (5.2). □
Proof of Theorem 1.2 For L\in {\mathcal{F}}_{i,o}^{n}, using (5.1) and Corollary 4.1, we have
taking K={\mathrm{\Lambda}}_{p,i}L in the inequality above, we immediately get inequality (1.22).
According to the condition of the equality that holds in inequalities (4.8) and (1.8), and noting that {\mathrm{\Lambda}}_{p,i}B=B and (1.15), we know with the equality in inequality (1.22) for p\ge 1 and 0<i<n1 if and only if K is a ball which is centered at the origin; for p\ge 1 and i=0 if and only if K is an ellipsoid which is centered at the origin. □
Proof of Theorem 1.3 Take L={\mathrm{\Gamma}}_{p,i}K in (5.2), we have
Using the Minkowski inequality (2.11) of the {L}_{p}mixed quermassintegrals, we have
Together with inequality (1.22), we can get
from this, we can get inequality (1.23).
According to the condition of the equality that holds in inequalities (2.11) and (1.22), we discuss the conditions of the equality that holds in (1.23) in the following four cases:

(1)
For the case p>1 and 0<i<n1, the equality holds in (1.23) if and only if K and {\mathrm{\Gamma}}_{p,i}{\mathrm{\Gamma}}_{p,i}K are dilates, and {\mathrm{\Gamma}}_{p,i}K is a ball which is centered at the origin. Because {\mathrm{\Gamma}}_{p,i}B=B, then {\mathrm{\Gamma}}_{p,i}K is a ball which is centered at the origin if and only if K is a ball which is centered at the origin. Therefore, K is a ball which is centered at the origin. While the equality {\mathrm{\Gamma}}_{p,i}B=B shows that {\mathrm{\Gamma}}_{p,i}K is a ball which is centered at the origin if and only if {\mathrm{\Gamma}}_{p,i}{\mathrm{\Gamma}}_{p,i}K is a ball which is centered at the origin. From this, for p>1 and 0<i<n1 the equality holds in inequality (1.23) if and only if K is a ball which is centered at the origin.

(2)
For the case p>1 and i=0, the equality holds in (1.23) if and only if K and {\mathrm{\Gamma}}_{p}{\mathrm{\Gamma}}_{p}K are dilates and {\mathrm{\Gamma}}_{p}K is an ellipsoid which is centered at the origin. Because {\mathrm{\Gamma}}_{p}K is an ellipsoid which is centered at the origin, and together with (1.10), K is an originsymmetric ellipsoid E if and only if {\mathrm{\Gamma}}_{p}K is an originsymmetric ellipsoid. On the other hand, the literature [11] tells us that if E is an ellipsoid which is centered at the origin, then {\mathrm{\Gamma}}_{p}E=E. From this, {\mathrm{\Gamma}}_{p}{\mathrm{\Gamma}}_{p}K is an ellipsoid which is centered at the origin. Therefore, for p>1 and i=0, the equality holds in inequality (1.23) if and only if K is an ellipsoid which is centered at the origin.

(3)
For the case p=1 and 0<i<n1, the equality holds in (1.23) if and only if K and {\mathrm{\Gamma}}_{1,i}{\mathrm{\Gamma}}_{1,i}K are homothetic, and {\mathrm{\Gamma}}_{1,i}K is a ball which is centered at the origin. From {\mathrm{\Gamma}}_{p,i}B=B, we know that {\mathrm{\Gamma}}_{1,i}B=B, then K is a ball which is centered at the origin. This {\mathrm{\Gamma}}_{1,i}B=B together with {\mathrm{\Gamma}}_{1,i}B=B, then {\mathrm{\Gamma}}_{1,i}{\mathrm{\Gamma}}_{1,i}K is a ball which is centered at the origin. Therefore, for p=1 and 0<i<n1, the equality holds in inequality (1.23) if and only if K is a ball which is centered at the origin.

(4)
For the case p=1 and i=0, the equality holds in (1.23) if K and \mathrm{\Gamma}{\mathrm{\Gamma}}_{1}K are homothetic, and {\mathrm{\Gamma}}_{1}K is an ellipsoid which is centered at the origin. Because {\mathrm{\Gamma}}_{1}K is an originsymmetric ellipsoid E if and only if K is an originsymmetric ellipsoid. On the other hand, from {\mathrm{\Gamma}}_{p}E=E, we know that {\mathrm{\Gamma}}_{1}K is an ellipsoid which is centered at the origin if and only if {\mathrm{\Gamma}}_{1}{\mathrm{\Gamma}}_{1}K is an ellipsoid which is centered at the origin. Therefore, for p=1 and i=0, the equality holds in inequality (1.23) if and only if K is an ellipsoid which is centered at the origin.
To sum up, the equality holds in (1.23) for p\ge 1 and 0<i<n1 if and only if K is a ball which is centered at the origin; for p\ge 1 and i=0 if and only if K is an ellipsoid which is centered at the origin. The proof is complete. □
References
Lutwak E: The BrunnMinkowskiFirey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 1993, 38: 131–150.
Lutwak E: The BrunnMinkowskiFirey theory II: affine and geominimal surface areas. Adv. Math. 1996, 118: 244–294. 10.1006/aima.1996.0022
Lutwak E: Mixed affine surface area. J. Math. Anal. Appl. 1987, 125: 351–360. 10.1016/0022247X(87)900977
Leichtweiß K: Bemerkungen zur definition einer erweiterten affinoberfläche von E. Lutwak. Manuscr. Math. 1989, 65: 181–197. 10.1007/BF01168298
Leichtweiß K: Affine Geometry of Convex Bodies. J. A. Barth, Heidelberg; 1998.
Lutwak E, Yang D, Zhang GY: {L}_{p}Affine isoperimetric inequalities. J. Differ. Geom. 2000, 56: 111–132.
Lutwak E: Mixed projection inequalities. Trans. Am. Math. Soc. 1985, 287: 91–106. 10.1090/S00029947198507662087
Wang, WD: Extremum problems in convex bodies geometry of {L}_{p}space. A dissertation submitted to Shanghai University for the degree of doctor in science 2, 113–115 (2006) (in Chinese)
Lü SJ, Leng GS: The {L}_{p}curvature images of convex bodies and {L}_{p}projection bodies. Proc. Indian Acad. Sci. Math. Sci. 2008, 118(3):413–424. 10.1007/s1204400800326
Yuan J, Lü SJ, Leng GS: The p affine surface area. Math. Inequal. Appl. 2007, 10(3):693–702.
Lutwak E, Zhang GY: BlaschkeSantal’o inequalities. J. Differ. Geom. 1997, 47: 1–16.
Lutwak E, Yang D, Zhang GY: {L}_{p}John ellipsoids. Proc. Lond. Math. Soc. 2005, 90: 497–520. 10.1112/S0024611504014996
Lin, S: Extremum problems in convex and discrete geometry. A dissertation submitted to Shanghai University for the degree of doctor in science (2006) (in Chinese)
Liu LJ, Wang W, He BW: Fourier transform and {L}_{p}mixed projection bodies. Bull. Korean Math. Soc. 2010, 47: 1011–1023. 10.4134/BKMS.2010.47.5.1011
Lu FH, Wang WD: Inequalities for {L}_{p}mixed curvature images. Acta Math. Sci. 2010, 30: 1044–1052. (in Chinese) 10.1016/S02529602(10)601014
Ma TY, Liu CY: The generalized BusemannPetty problem for dual {L}_{p}mixed centroid bodies. J. Southwest Univ., Nat. Sci. Ed. 2012, 34(4):105–112. (in Chinese)
Ma TY, Liu CY: The generalized Shephard problem for {L}_{p}mixed projection bodies and MinkowskiFunk transforms. J. Shandong Univ. (Nat. Sci.) 2012, 47(10):21–30. (in Chinese)
Wang WD, Leng GS: The Petty projection inequality for {L}_{p}mixed projection bodies. Acta Math. Sin. Engl. Ser. 2007, 23(8):1485–1494. (in Chinese) 10.1007/s1011400508949
Ma TY: On {L}_{p}mixed centroid bodies and dual {L}_{p}mixed centroid bodies. Acta Math. Sin., Chin. Ser. 2010, 53(2):301–314. (in Chinese)
Gardner RJ: Geometric Tomography. Cambridge University Press, Cambridge; 1995.
Schneider R: Convex Bodies: The BrunnMinkowski Theory. Cambridge University Press, Cambridge; 1993.
Firey WJ: p Means of convex bodies. Math. Scand. 1962, 10: 17–24.
Firey WJ: Polar means of convex bodies and a dual to the BrunnMinkowski theorem. Can. J. Math. 1961, 13: 444–453. 10.4153/CJM19610370
Firey WJ: Mean crosssection measures of harmonic means of convex bodies. Pac. J. Math. 1961, 11: 1263–1266. 10.2140/pjm.1961.11.1263
Wang WD, Leng GS: {L}_{p}Dual mixed quermassintegrals. Indian J. Pure Appl. Math. 2005, 36(4):177–188.
Wang WD, Leng GS: {L}_{p}Mixed affine surface area. J. Math. Anal. Appl. 2007, 54: 1–14.
Lutwak E: Dual mixed volumes. Pac. J. Math. 1975, 58(2):531–538. 10.2140/pjm.1975.58.531
Petty CM: Isoperimetric problems. In Proc.Conf. Convexity and Combinatorial Geometry. University of Oklahoma, Oklahoma; 1972:26–41. Univ. Oklahoma, 1971
Wang WD, Wei DJ, Xiang Y: Some inequalities for the {L}_{p}curvature image. J. Inequal. Appl. 2009., 2009: Article ID 320786 10.1155/2009/320786
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The author is supported by the NNSF of China (11161019, 11371224).
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Ma, TY. Some inequalities related to (i,j)type {L}_{p}mixed affine surface area and {L}_{p}mixed curvature image. J Inequal Appl 2013, 470 (2013). https://doi.org/10.1186/1029242X2013470
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DOI: https://doi.org/10.1186/1029242X2013470