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Inequalities of Aleksandrov body
Journal of Inequalities and Applications volume 2011, Article number: 39 (2011)
Abstract
A new concept of p Aleksandrov body is firstly introduced. In this paper, p BrunnMinkowski inequality and p Minkowski inequality on the p Aleksandrov body are established. Furthermore, some pertinent results concerning the Aleksandrov body and the p Aleksandrov body are presented.
2000 Mathematics Subject Classification:
52A20 52A40
1 Introduction
The notion of Aleksandrov body was firstly introduced by Aleksandrov to solve Minkowski problem in 1930s in [1]. The Aleksandrov body establishes the relationship between the convex body containing the origin and the positive continuous functions and characterizes the convex body by means of the positive continuous functions. The Aleksandrov body not only be used to solve Minkowski problem but also has a wide range of applications in other areas of Convex Geometric Analysis. Then, the Aleksandrov body is an essential matter in the BrunnMinkowski theory and plays an important role in Convex Geometric Analysis. In recent years, Ball [2], Gardner [3, 4], Lutwak [5–10], Klain [11], Hug [12], Haberl [13], Schneider [14], Stancu [15], Umanskiy [16] and Zhang [17] have given considerable attention to the BrunnMinkowski theory and their various generalizations.
The purpose of this paper is to study comprehensively the Aleksandrov body, and most importantly, the L_{ p } analogues of Aleksandrov body become a major goal. Here, a new geometric body is firstly introduced, called p Aleksandrov body. Meanwhile, p BrunnMinkowski inequality and p Minkowski inequality for the p Aleksandrov bodies associated with positive continuous functions are established. Furthermore, some related results, including of the uniqueness results, the convergence results for the Aleksandrov bodies and the p Aleksandrov bodies associated with positive continuous functions, are presented.
Let ${\mathcal{K}}^{n}$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space ℝ ^{n} , ${\mathcal{K}}_{0}^{n}$ denote the set of convex bodies containing the origin in their interiors. Let V (K) denote the n dimensional volume of body K, for the standard unit ball B in ℝ ^{n} , denote ω_{ n } = V (B), and let S^{n 1}denote the unit sphere in ℝ ^{n} .
Let C^{+}(S^{n 1}) denote the set of positive continuous functions on S^{n 1}, endowed with the topology derived from the max norm. Given a function f ∈C^{+}(S^{n 1}), the set
has a unique maximal element, then we denoted the Aleksandrov body associated with the function f ∈C^{+}(S^{n 1}) by
The volume of body K(f) is denoted by V (K(f)). Following Aleksandrov (see [18]), define the volume V (f) of a function f as the volume of the Aleksandrov body associated with the positive continuous function f.
In this paper, we generalize and improve BrunnMinkowski inequality and Minkowski inequality for the Aleksandrov bodies associated with positive continuous functions and establish p Minkowski inequality and p BrunnMinkowski inequality for the Aleksandrov bodies and the p Aleksandrov bodies associated with positive continuous functions as follows.
Theorem 1
If$Q\in {\mathcal{K}}_{0}^{n}$, f ∈C^{+}(S^{n 1}), and p ≥ 1, then
with equality if and only if there exists a constant c > 0 such that h_{ Q } = cf, almost everywhere with respect to S(Q, ·) on S^{n 1}.
Theorem 2
If p ≥ 1, f, g ∈C^{+}(S^{n 1}), and λ, μ ∈ℝ^{+}, then
with equality if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
The other aim of this paper is to establish the following inequality for the Aleksandrov bodies and the p Aleksandrov bodies associated with positive continuous functions.
Theorem 3
If K(f ), $K\left(g\right)\in {\mathcal{K}}_{e}^{n}$, are the Aleksandrov bodies associated with the functions f, g ∈C^{+}(S^{n 1}), and n ≠ p ≥ 1, then
with equality if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
More interrelated notations, definitions, and their background materials are exhibited in the next section.
2 Definition and notation
The setting for this paper is ndimensional Euclidean space ℝ ^{n} . Let ${\mathcal{K}}^{n}$ denote the set of convex bodies (compact, convex subsets with nonempty interiors), ${\mathcal{K}}_{0}^{n}$ denote the subset of ${\mathcal{K}}^{n}$ that contains the origin in their interiors, and ${\mathcal{K}}_{e}^{n}$ denote the subset of ${\mathcal{K}}^{n}$ that are centered in ℝ ^{n} . We reserve the letter u for unit vector and the letter B for the unit ball centered at the origin. The surface of B is S^{n 1}, and the volume of B denotes ω_{ n } .
For φ ∈GL(n), let φ^{t}, φ^{1}, and φ ^{t} , denote the transpose, inverse, and inverse of the transpose of φ, respectively. If K ∈ ${\mathcal{K}}^{n}$, the support function of K, h_{ K } = h(K, ·): ℝ ^{n} → (0, ∞), is defined by
where u · x denotes the standard inner product of u and x.
The set ${\mathcal{K}}^{n}$ will be viewed as equipped with the usual Hausdorff metric, d, defined by d(K, L) = h_{ K }  h_{ L } _{∞}, where  · _{∞} is the sup (or max) norm on the space of continuous functions on the unit sphere, C(S^{n 1}).
For K, L ∈ ${\mathcal{K}}^{n}$, and α, β ≥ 0 (not both zero), the Minkowski linear combination, αK + βL ∈ ${\mathcal{K}}^{n}$ is defined by
Firey introduced, for each real p ≥ 1, new linear combinations of convex bodies: For K, $L\in {\mathcal{K}}_{0}^{n}$, and α, β ≥ 0 (not both zero), the Firey combination, $\alpha \cdot K{+}_{p}\beta \cdot L\in {\mathcal{K}}_{0}^{n}$ whose support function is defined by (see [19])
Obviously, α · K = α^{1/p}K.
For K, L ∈ ${\mathcal{K}}^{n}$, and α, β ≥ 0 (not both zero), by the Minkowski existence theorem (see [3, 14]), there exists a convex body α ⋅ K + β ⋅ L ∈ ${\mathcal{K}}^{n}$, such that
where S(K, ·) denotes the surface area measure of K, and the linear combination α · K + β · L is called a Blaschke linear combination.
Lutwak generalized the notion of Blaschke linear combination in [5]:
For K, $L\in {\mathcal{K}}_{e}^{n}$, and n ≠ p ≥ 1, define $K{+}_{p}L\in {\mathcal{K}}_{e}^{n}$ by
The existence and uniqueness of K + _{ p } L are guaranteed by Minkowski's existence theorem in [5].
2.1 Mixed volume and p mixed volume
If K_{ i } ∈ ${\mathcal{K}}^{n}$ (I = 1, 2, ..., r) and λ_{ i } (i = 1, 2,..., r) are nonnegative real numbers, then of fundamental importance is the fact that the volume of ${\sum}_{i=1}^{r}{\lambda}_{i}{K}_{i}$ is a homogeneous polynomial in λ _{ i } given by
where the sum is taken over all n tuples (i_{1},... i_{ n } ) of positive integers not exceeding r. The coefficient $V\left({K}_{{i}_{1}}\dots {K}_{{i}_{n}}\right)$, which is called the mixed volume of ${K}_{{i}_{1}}\dots {K}_{{i}_{n}}$, depends only on the bodies ${K}_{{i}_{1}}\dots {K}_{{i}_{n}}$ and is uniquely determined by (2.5). If K_{1} = ... K_{ni}= K and K_{n  i+1}= ... = K_{ n } = L, then the mixed volume V (K_{1} ... K_{ n } ) is usually written as V_{ i } (K, L).
Let r = 1 in (2.5), we see that
Further, from (2.5), it follows immediately that
Aleksandrov (see [1]) and Fenchel and Jessen (see [20]) have shown that corresponding to each K ∈ ${\mathcal{K}}^{n}$, there is a positive Borel measure, S(K, ·) on S^{n 1}, called the surface area measure of K, such that
for all Q ∈ ${\mathcal{K}}^{n}$.
For p ≥ 1, the p mixed volume V_{ p } (K, L) of K, $L\in {\mathcal{K}}_{0}^{n}$, was defined by (see [5])
That the existence of this limit was demonstrated in [5].
It was also shown in [5], that corresponding to each $K\in {\mathcal{K}}_{0}^{n}$, there is a positive Borel measure, S_{ p } (K, ·) on S^{n  1}such that
for all $Q\in {\mathcal{K}}_{0}^{n}$. It turns out that the measure S_{ p } (K, ·) is absolutely continuous with respect to S(K, ·) and has RadonNikodym derivative,
From (2.7) and (2.8), we have
where S(K, ·) = S_{0}(K, ·) is the surface area measure of K.
Obviously, for each $K\in {\mathcal{K}}_{0}^{n}$, p ≥ 1,
2.2 Aleksandrov body
If a function f ∈ C^{+}(S^{n 1}) (denoted the set of positive continuous functions on S^{n 1}and endowed with the topology derived from the max norm), the set
has a unique maximal element, then the Aleksandrov body associated with the function f is denoted by
From (2.11) and (2.1), we have: If f, g ∈ C+(S^{n 1}), and λ, μ ≥ 0 (not both zero), then
Obviously, if f is the support function of a convex body K, then the Aleksandrov body associated with f is K.V (K(f )) denotes the volume of body K(f ). Following Aleksandrov (see [18]), define the volume V (f ) of a function f as the volume of the Aleksandrov body associated with the positive function f.
For $Q\in {\mathcal{K}}_{0}^{n}$, f ∈ C^{+}(S^{n 1}), and p ≥ 1, V_{ p } (Q, f ) is defined by (see [5])
Obviously, V_{ p } (K, h_{ K } ) = V (K), for all $K\in {\mathcal{K}}_{0}^{n}$.
2.3 p Aleksandrov body
Definition 1
Let f, g ∈ C^{+}(S^{n 1}), p ≥ 1, and
define
Then, the set
has a unique maximal element.
We denote the pAleksandrov body associated with the function f + _{ p } g ∈ C^{+} (S^{n 1}) by
for p ≥ 1.
The volume of body K_{ p } (f + _{ p } g) is denoted by V (K_{ p } (f + _{ p } g)), and define the volume V (f + _{ p } g) of the function f + _{ p } g as the volume of the p Aleksandrov body associated with the positive function f + _{ p } g.
From (2.2), we have the following result: If f, g ∈ C^{+}(S^{n 1}), and p ≥ 1, then
We note that the equality condition in (2.16) is clearly holds, when f and g are the support functions of K(f ) and K(g), respectively. Also, the case p = 1 of (2.16) is just (2.12).
3 Proof of the main results
The following Lemmas will be required to prove our main theorems.
Lemma 1
[5]If K(f ) is the Aleksandrov body associated with f ∈ C^{+}(S^{n 1}), then h_{K(f)}= f almost everywhere with respect to the measure S(K(f ), ·) on S^{n 1}.
Obviously, if K(f ) is the Aleksandrov body corresponding to a given function f ∈ C^{+}(S^{n 1}), its support function has the property that 0 < h_{ K } ≤ f and V (f ) = V (h_{K(f )}).
Lemma 2
[5]If p ≥ 1, K(f ) is the Aleksandrov body associated with f ∈ C^{+}(S^{n 1}), then V (f ) = V (K(f )) = V_{ p } (K(f ), f ), i.e.
Lemma 3
[5]If$K\in {\mathcal{K}}_{0}^{n}$, f ∈ C^{+}(S^{n 1}), then, for p ≥ 1,
We get the following BrunnMinkowski inequality for the Aleksandrov bodies associated with positive continuous functions.
Lemma 4
If f, g ∈ C^{+}(S^{n 1}), and λ, μ ∈ℝ^{+}, then
with equality if and only if there exist a constant c > 0 and t ≥ 0, such that f = cg + t, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
Proof
Since f, g ∈ C^{+}(S^{n 1}), from (2.11), (2.12) and the BrunnMinkowski inequality (see [21]), we get
The equality condition in (3.3) is that f, g are the support functions of K(f ) and K(g) which are homothetic, respectively.
From Lemma 1 and Lemma 2, we get the following result
with equality if and only if there exist a constant c > 0 and t ≥ 0, such that f = cg + t, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
An immediate consequence of the definition of a Firey linear combination, and the integral representation (2.13), is that for $Q\in {\mathcal{K}}_{0}^{n}$, the p mixed volume
is Firey linear.
Lemma 5
If p ≥ 1, $Q\in {\mathcal{K}}_{0}^{n}$, f, g ∈ C^{+}(S^{n 1}), and λ, μ ∈ ℝ^{+}, then
Proof
From (2.13), (2.14), we obtain
In the following, we will prove the p Minkowski inequality for the Aleksandrov bodies associated with positive continuous functions.
Proof of Theorem 1.
Firstly, let p = 1 in Lemma 3, we get
let $\epsilon =\frac{t}{1t}$, we have
Let
we see that
From Lemma 4, we know that f is concave, i.e.
Thus,
According to the equality condition in inequality (3.3), and using Lemma 1 and Lemma 2, we have the equality holds in inequality (3.6), if and only if there exist a constant c > 0 and t ≥ 0, such that h_{ Q } = cf + t, almost everywhere with respect to S(Q, ·) on S^{n 1}.
Secondly, from the Hölder inequality (see [22]), together with the integral representations (2.13) and (2.6), we obtain
when this combined with inequality (3.6), we have
To obtain the equality conditions, we note that there is equality in Hölder's inequality precisely when V_{1}(Q, f )h_{ Q } = V (Q)f, almost everywhere with respect to the measure S(Q, ·) on S^{n 1}. Combining the equality conditions in (3.6), and using Lemma 1, it shows that the equality holds if and only if there exists a constant c > 0 such that h_{ Q } = cf, almost everywhere with respect to S(Q, ·) on S^{n 1}.
Using the above Lemmas and Theorem 1, we can get the following Corollaries describing the uniqueness results.
Corollary 1
Suppose K, $L\in {\mathcal{K}}_{0}^{n}$, and$\mathcal{F}$ ⊂ C^{+} (S^{n 1}) is a class of functions such that h_{ K }, h_{ L } ∈ $\mathcal{F}$. (i) If n ≠ p > 1, and V_{ p } (K, f ) = V_{ p } (L, f ), for all f ∈ $\mathcal{F}$, then K = L. (ii) If p = n, and V_{ p } (K, f ) ≥ V_{ p } (L, f ), for all f ∈ $\mathcal{F}$, then K and L are dilates, and hence
Proof
If n ≠ p > 1, take f = h_{ K } , and from (2.13), Lemma 2 and Theorem 1, we get
Hence,
Similarly, take f = h_{ L } , we get
In view of the equality conditions of Theorem 1, we obtain that K = L.
If n = p, the hypothesis together with Theorem 1, we have
with equality in the right inequality implying that L and K(f ) are dilates. Take f = h_{ K } , since n = p, the terms on the left and right are identical, and thus, K and L must dilates; hence,
Corollary 2
Suppose f, g ∈ C^{+}(S^{n 1}), and$\mathcal{F}$ ⊂ C^{+} (S^{n 1}) is a class of functions such that f, g ∈ $\mathcal{F}$. If p > 1, and
then f = g almost everywhere on S^{n 1}.
Proof
Since f, g ∈ C^{+}(S^{n 1}), according to (2.11), we denote two Aleksandrov bodies K(f ) and K(g). From the hypothesis, taking Q = K(f ), and using Lemma 2 and Theorem 1, we get
then,
Similarly, take Q = K(g), we get
From the equality conditions of Theorem 1, we obtain
In view of the definition of Aleksandrov body, and using Lemma 1, then
Corollary 3
Suppose n ≠ p > 1, and f, g ∈ C^{+}(S^{n 1}), such that S_{ p } (K(f ), ·) ≤ S_{ p } (K(g), ·).

(i)
If V (f ) ≥ V (g), and p < n, then f = g almost everywhere on S^{n 1}.

(ii)
If V (f ) ≤ V (g), and p > n, then f = g almost everywhere on S ^{n 1}.
Proof
Suppose a function φ ∈ C^{+}(S^{n 1}), and n ≠ p > 1, since S_{ p } (K(f ), ·) ≤ S_{ p } (K(g), ·), it follows from the integral representation (2.13) and (2.8) that
As before, take φ = h_{K(g)}, from Lemma 1, Lemma 2, and Theorem 1, we get
Applying the hypothesis, and from the definition of the Aleksandrov body and Lemma 1, we obtain the desired results.
Corollary 4
Suppose n ≠ p ≥ 1, f, g ∈ C^{+}(S^{n 1}), and$\mathcal{F}$ ⊂ C^{+} (S^{n 1}) is a class of functions such that f, g ∈ $\mathcal{F}$. If
then f = g almost everywhere on S^{n 1}.
Proof
According to (2.11), we denote two Aleksandrov bodies K(f ) and K(g). From the hypothesis, taking K = K(f ) and K = K(g), and combining with Lemma 2 and Theorem 1, respectively, we obtain
Hence, in view of the equality conditions of Theorem 1, the definition of Aleksandrov body, and Lemma 1, we get the desired result
Now, the p BrunnMinkowski inequality for the p Aleksandrov bodies and the Aleksandrov bodies associated with positive continuous functions is established as following.
Proof of Theorem 2.
From Lemma 5 and Theorem 1, we get
with equality if and only if K(f) and K(g) are dilates of Q.
Now, take Q = K_{ p } (λ · f + _{ p } μ · g), use (2.10), and recall V (f ) = V (K(f )) = V_{ p } (K(f ), f ), we have
Also, we note that the equality holds, if and only if K(f ) and K(g) are dilates. Using Lemma 1, we get the condition of equality holds if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
Then, we will prove Theorem 3 by using the generalized Blaschke linear combination.
Proof of Theorem 3.
Suppose a function φ ∈ C^{+}(S^{n 1}), and n ≠ p ≥ 1, from the integral representation (2.13), (2.8), and (2.4), it follows that for K(f ), $K\left(g\right)\in {\mathcal{K}}_{e}^{n}$,
which together with Theorem 1, yields
with equality if and only if K(f); K(g) and K(φ ) are dilates.
Now, take $\varphi ={h}_{K\left(f\right){+}_{p}K\left(g\right)}$, recall Vp(K, h_{ K } ) = V (K ), and from Lemma 2, we get
In view of (2.16), we have
Hence, we get
From Lemma 2 again, we obtain
In view of the equality condition (3.9), and from Lemma 1, we get the equality holds if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
Remark 1
The case p = 1 of the inequality of Theorem 3 is
with equality if and only if there exists a constant c > 0 such that f = cg, almost everywhere with respect to S(K(f ), ·) on S^{n 1}.
The above inequality (3.10) is just the KneserSüss inequality type for the Aleksandrov bodies associated with positive continuous functions.
Actually, from these above proofs, we see BrunnMinkowski inequality, Minkowski inequality, and KnesserSüss inequality are equivalent.
4 Convergence of Aleksandrov body
In this section, we establish a convergent result about the Aleksandrov bodies associated with positive continuous functions.
The following Lemmas will be required to prove our main result.
Lemma 6
[7]If p ≥ 1, and K_{ i } is a sequence of bodies in${\mathcal{K}}_{0}^{n}$, such that ${K}_{i}\to {K}_{0}\in {\mathcal{K}}_{0}^{n}$, then S_{ p } (K_{ i } , ·) → S_{ p } (K_{0}, ·), weakly.
Lemma 7
Suppose${K}_{i}\to K\in {\mathcal{K}}_{0}^{n}$, and f_{ i } → f ∈ C^{+}(S^{n 1}). If p ≥ 1, then V_{ p } (K_{ i }, f_{ i } ) → V_{ p } (K, f ).
Proof
Since f_{ i } → f ∈ C^{+}(S^{n 1}), the f_{ i } are uniformly bounded on S^{n 1}. Hence,
By Lemma 6, K_{ i } → K implies that
Hence,
In view of the integral representation (2.13) and (2.8), we get the desired result. The convergence result will be established as following.
Theorem 4
Suppose p > 1, f ∈ C^{+}(S^{n 1}). If f_{ i } is a sequence of functions in C^{+}(S^{n 1}), such that
then f_{ i } → f .
Proof
Firstly, since f_{ i } is a sequence in C^{+}(S^{n 1}), f_{ i } are uniformly bounded on S^{n 1}.
Applying the Blaschke selection theorem (see [3]), it guarantees the existence of a subsequence of the f_{ i } , which is again denoted by f_{ i } , converging to a positive continuous function f_{0} on S^{n}^{1}.
Secondly, since f_{ i } are uniformly bounded on S^{n1},
Define ${\stackrel{\u0304}{f}}_{i}\in {C}^{+}\left({S}^{n1}\right)$, by ${\stackrel{\u0304}{f}}_{i}=1{+}_{p}{f}_{i}$. Since ${\stackrel{\u0304}{f}}_{i}\to {\stackrel{\u0304}{f}}_{0}$, it follows from Lemma 7 that
However, since V_{ p } (Q, f_{ i } ) → V_{ p } (Q, f ), for all $Q\in {\mathcal{K}}_{0}^{n}$, and ${\stackrel{\u0304}{f}}_{i}=1{+}_{p}{f}_{i}$, for all i > 0, it follows from Lemma 5 that
Thus,
By Corollary 2, this means ${\stackrel{\u0304}{f}}_{0}=1{+}_{p}f$, which shows f_{0} = f .
Thus, every subsequence of f_{ i } has a subsequence that converges to f .
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Acknowledgements
The authors express their deep gratitude to the referees for their many very valuable suggestions and comments. The research of HuYan and JiangJunhua was supported by National Natural Science Foundation of China (10971128), Shanghai Leading Academic Discipline Project (S30104), and the research of HuYan was partially supported by Innovation Program of Shanghai Municipal Education Commission (10yz160).
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HY and JJH jointly contributed to the main results Theorem 1, Theorem 2, Theorem 3 and Theorem 4. HY drafted the manuscript and made the text file. Both authors read and approved the final manuscript.
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Yan, H., Junhua, J. Inequalities of Aleksandrov body. J Inequal Appl 2011, 39 (2011). https://doi.org/10.1186/1029242X201139
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Keywords
 Aleksandrov body
 p Aleksandrov body
 BrunnMinkowski inequality
 Minkowski inequality