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On p-radial Blaschke and harmonic Blaschke additions
Journal of Inequalities and Applications volume 2017, Article number: 308 (2017)
Abstract
In the paper, we first improve the radial Blaschke and harmonic Blaschke additions and introduce the p-radial Blaschke and p-harmonic Blaschke additions. Following this, Dresher type inequalities for the radial Blaschke-Minkowski homomorphisms with respect to p-radial Blaschke and p-harmonic Blaschke additions are established.
1 Notation and preliminaries
The setting for this paper is an n-dimensional Euclidean space \({\Bbb {R}}^{n}\). We reserve the letter u for unit vectors, and the letter B is reserved for the unit ball centered at the origin. The surface of B is \(S^{n-1}\). The volume of the unit n-ball is denoted by \(\omega_{n}\). We use \(V(K)\) for the n-dimensional volume of a body K. Associated with a compact subset K of \({\Bbb {R}}^{n}\), which is star-shaped with respect to the origin, is its radial function \(\rho(K,\cdot): S^{n-1}\rightarrow{\Bbb {R}}\) defined for \(u\in S^{n-1}\) by
If \(\rho(K,\cdot)\) is positive and continuous, K will be called a star body. Let \({\mathcal {S}}^{n}\) denote the set of star bodies in \({\Bbb {R}}^{n}\). Let δ̃ denote the radial Hausdorff metric, i.e., if \(K, L\in{\mathcal {S}}^{n}\), then \(\tilde{\delta}(K,L)=|\rho(K,u)-\rho(L,u)|_{\infty}\), where \(|\cdot|_{\infty}\) denotes the sup-norm on the space of continuous functions \(C(S^{n-1})\).
1.1 Dual mixed volumes
The radial Minkowski linear combination, \(\lambda_{1}K_{1}\widetilde {+}\cdots\widetilde{+}\lambda_{r} K_{r}\) is defined by
for \(K_{1},\ldots,K_{r}\in{\mathcal {S}}^{n}\) and \(\lambda_{1},\ldots,\lambda_{r}\in{\Bbb {R}}\). It has the following important property (see [1]):
for \(K, L\in{\mathcal {S}}^{n}\) and \(\lambda, \mu\geq0\). For \(K_{1},\ldots,K_{r}\in{\mathcal {S}}^{n}\) and \(\lambda_{1},\ldots,\lambda_{r}\geq0\), the volume of the radial Minkowski linear combination \(\lambda_{1}K_{1}\widetilde{+}\cdots\widetilde{+}\lambda_{r}K_{r}\) is a homogeneous polynomial of degree n in the \(\lambda_{i}\),
If we require the coefficients of the polynomial in (1.3) to be symmetric in their arguments, then they are uniquely determined. The coefficient \(\widetilde{V}(K_{i_{1}},\ldots,K_{i_{n}})\) is nonnegative and depends only on the bodies \(K_{i_{1}},\ldots,K_{i_{n}}\). It is called the dual mixed volume of \(K_{i_{1}},\ldots,K_{i_{n}}\).
If \(K_{1},\ldots,K_{n}\in{\mathcal {S}}^{n}\), then the dual mixed volume \(\widetilde{V}(K_{1},\ldots,K_{n})\) can be represented in the form (see [2])
If \(K_{1}=\cdots=K_{n-i}=K\), \(K_{n-i+1}=\cdots=K_{n}=L\), then the dual mixed volume is written as \(\widetilde{V}_{i}(K,L)\). If \(L=B\), then the dual mixed volume \(\widetilde{V}_{i}(K,L)=\widetilde{V}_{i}(K,B)\) is written as \(\widetilde{W}_{i}(K)\). For \(K,L\in{\mathcal {S}}^{n}\), the ith dual mixed volume of K and L, \(\widetilde{V}_{i}(K,L)\) can be extended to all \(i\in{\Bbb {R}}\) by
where \(i\in{\Bbb {R}}\). Thus, if \(K\in{\mathcal {S}}^{n}\), then
1.2 Mixed intersection bodies
For \(K\in{\mathcal {S}}^{n}\), there is a unique star body I K whose radial function satisfies, for \(u\in S^{n-1}\),
where v is \((n-1)\)-dimensional dual volume. It is called the intersection body of K. The volume of the intersection body of K is given by (see [1])
The mixed intersection body of \(K_{1},\ldots,K_{n-1}\in{\mathcal {S} }^{n}\), denoted by \(\mathbf{I}(K_{1},\ldots,K_{n-1})\), is defined by
where á¹½ is the \((n-1)\)-dimensional dual mixed volume. If \(K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=L\), then \({\bf I}(K_{1},\ldots,K_{n-1})\) is written as \(\mathbf{I}_{i}(K,L)\). If \(L=B\), then \(\mathbf{I}_{i}(K,L)\) is written as \(\mathbf{I}_{i}K\) and called the ith intersection body of K. For \(\mathbf{I}_{0}K\), we simply write I K.
2 Improvement of the radial Blaschke addition
Let us recall the concept of radial Blaschke addition defined by Lutwak [1]. Suppose that K and L are star bodies in \({\Bbb {R}}^{n}\), the radial Blaschke addition denoted by \(K\widehat{+}L\) is a star body whose radial function is
The dual Knesser-Süss inequality for the radial Blaschke addition was established by Lutwak [1]. If \(K,L\in{\mathcal {S}}^{n}\), then
with equality if and only if K and L are dilates.
In the section, we give a generalized concept of the radial Blaschke addition.
Definition 2.1
If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(\lambda, \mu>0\) (not both zero), the p-radial Blaschke linear combination of K and L denoted by \(\lambda\diamond K\widehat{+}_{p}\mu\diamond L\) is a star body whose radial function is defined by
From (2.3), it is easy to see that
When \(\lambda=\mu=1\), the p-radial Blaschke combination becomes the p-radial Blaschke addition \(K\widehat{+}_{p}L\) and
Obviously, when \(p=0\), (2.4) becomes (2.1).
In the following, we define the dual mixed quermassintegral with respect to the p-radial Blaschke addition. First, we show two propositions. The following proposition follows immediately from (2.3) with L’Hôpital’s rule.
Proposition 2.2
Let \(0\leq p< n-1\), \(0\leq i< n\) and \(\varepsilon>0\). If \(K,L\in{\mathcal {S}}^{n}\), then
The following proposition follows immediately from Proposition 2.2 and (1.6).
Proposition 2.3
Let \(0\leq p< n-1\), \(0\leq i< n\) and \(\varepsilon>0\). If \(K,L\in{\mathcal {S}}^{n}\), then
Definition 2.4
For \(0\leq p< n-1\), \(0\leq i< n\) and \(K,L\in {\mathcal {S}}^{n}\), the p-dual mixed quermassintegral of star bodies K and L, denoted by \(\widetilde{W}_{p,i}(K,L)\), is defined by
Obviously, when \(K=L\), \(\widetilde{W}_{p,i}(K,L)\) becomes the dual quermassintegral of star body K, i.e., \(\widetilde{W}_{p,i}(K,K)=\widetilde{W}_{i}(K)\). Taking \(i=0\) in (2.7), \(\widetilde{W}_{p,i}(K,L)\) becomes the p-dual mixed volume \(\widetilde{V}_{p}(K,L)\) and
From (2.7), combining Hölder’s integral inequality (see [3]) gives the following.
Proposition 2.5
(Minkowski type inequality)
If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\) and \(0\leq p< n-1\), then
with equality if and only if K and L are dilates.
Taking \(i=0\) in (2.9), we have: If \(K,L\in{\mathcal {S}}^{n}\) and \(0\leq p< n-1\), then
with equality if and only if K and L are dilates. In the following, we establish the Brunn-Minkowski inequality for the p-radial Blaschke addition.
Proposition 2.6
If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\) and \(0\leq p< n-1\), then
with equality if and only if K and L are dilates.
Proof
From (2.3) and (2.7), it is easily seen that the p-dual mixed quermassintegral \(\widehat{W}_{p,i}(K,L)\) is linear with respect to the p-radial Blaschke addition and together with inequality (2.9) shows that
with equality if and only if K and L are dilates of Q. Take \(K\widehat{+}_{p}L\) for Q in (2.12), recall that \(\widetilde{W}_{p,i}(Q,Q)=\widetilde{W}_{i}(Q)\), inequality (2.11) follows easy.
Taking \(i=0\) in (2.11), we obtain that if \(K,L\in{\mathcal {S}}^{n}\) and \(0\leq p< n-1\), then
with equality if and only if K and L are dilates. Taking \(p=0\) and \(i=0\) in (2.11), (2.11) becomes the well-known dual Knesser-Süss inequality (2.2). □
3 Improvement of the harmonic Blaschke addition
Let us recall the concept of harmonic Blaschke addition defined by Lutwak [4]. Suppose that K and L are star bodies in \({\Bbb {R}}^{n}\), the harmonic Blaschke addition denoted by \(K\breve{+}L\) is defined by
Lutwak’s Brunn-Minkowski inequality for the harmonic Blaschke addition was established (see [4]). If \(K,L\in{\mathcal {S}}^{n}\), then
with equality if and only if K and L are dilates.
In the section, we give an improved concept of the harmonic Blaschke addition.
Definition 3.1
For \(0\leq i< n\), \(p< i-1\) and \(K,L\in{\mathcal {S}}^{n}\), we define the p-harmonic Blaschke addition of star bodies K and L denoted by \(K\breve{+}_{p}L\) and defined by
The Brunn-Minkowski inequality for the p-harmonic Blaschke addition follows immediately from (1.6), (3.3) and Minkowski’s integral inequality (see [3]).
Proposition 3.2
If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\) and \(p< i-1\), then
with equality if and only if K and L are dilates.
4 Radial Blaschke-Minkowski homomorphisms
Definition 4.1
([5])
A map \(\Psi: {\mathcal {S}}^{n}\rightarrow {\mathcal {S}}^{n}\) is called a radial Blaschke-Minkowski homomorphism if it satisfies the following conditions:
-
(a)
Ψ is continuous.
-
(b)
For all \(K,L\in{\mathcal {S}}^{n}\),
$$\Psi(K\ddot{+}L)=\Psi(K)\widetilde{+}\Psi(L). $$ -
(c)
For all \(K,L\in{\mathcal {S}}^{n}\) and every \(\vartheta\in SO(n)\),
$$\Psi(\vartheta K)=\vartheta\Psi(K), $$where \(SO(n)\) is the group of rotations in n dimensions.
Radial Blaschke-Minkowski homomorphisms are important examples of star body valued valuations. Their natural duals, Blaschke-Minkowski homomorphisms, are an important notion in the theory of convex body valued valuations (see, e.g., [6–12] and [13–20]). In 2006, Schuster [5] established the following Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms of star bodies. If K and L are star bodies in \({\Bbb {R}}^{n}\), then
with equality if and only if K and L are dilates.
If K and L are star bodies in \({\Bbb {R}}^{n}\), \(p\neq0\) and \(\lambda, \mu\geq0\), then \(\lambda\cdot K\widetilde{+}_{p}\mu\cdot L\) is the star body whose radial function is given by (see, e.g., [21])
The addition \(\widetilde{+}_{p}\) is called \(L_{p}\)-radial addition. The \(L_{p}\) dual Brunn-Minkowski inequality states: If \(K,L\in{\mathcal {S}}^{n}\) and \(0< p\leq n\), then
with equality when \(p\neq n\) if and only if K and L are dilates. The inequality is reversed when \(p>n\) or \(p<0\) (see [21]).
In 2013, an \(L_{p}\) Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms was established in [22]: If K and L are star bodies in \({\Bbb {R}}^{n}\) and \(0< p< n-1\), then
with equality if and only if K and L are dilates. Taking \(p=1\), (4.3) reduces to (4.1).
Theorem 4.2
(see [5])
Let \(\Psi: {\mathcal {S}}^{n}\rightarrow{\mathcal {S}}^{n}\) be a radial Blaschke-Minkowski homomorphism. There is a continuous operator \(\Psi: \underbrace{{{\mathcal {S}}^{n}\times\cdots\times\mathcal {S}}^{n}}_{n-1}\rightarrow{\mathcal {S}}^{n}\) symmetric in its arguments such that, for \(K_{1},\ldots,K_{m}\in{\mathcal {S}}^{n}\) and \(\lambda_{1},\ldots,\lambda_{m}\geq0\),
Clearly, Theorem 4.2 generalizes the notion of radial Blaschke-Minkowski homomorphisms. We call \(\Psi: {\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}\rightarrow{\mathcal {S}}^{n}\) a mixed radial Blaschke-Minkowski homomorphism induced by Ψ. Mixed radial Blaschke-Minkowski homomorphisms were first studied in more detail in [23, 24]. If \(K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=L\), we write \(\Psi_{i}(K,L)\) for \(\Psi(\underbrace{K,\ldots,K}_{n-i-1},\underbrace{L,\ldots,L}_{i})\). If \(K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=B\), we write \(\Psi_{i} K\) for \(\Psi(\underbrace{K,\ldots,K}_{n-i-1},\underbrace{B,\ldots,B}_{i})\) and call \(\Psi_{i} K\) the mixed Blaschke-Minkowski homomorphism of order i of K. \(\Psi_{0} K\) is written simply as ΨK.
Lemma 4.3
(see [5])
A map \(\Psi: {\mathcal {S}}^{n}\rightarrow{\mathcal {S}}^{n}\) is a radial Blaschke-Minkowski homomorphism if and only if there is a measure \(\mu\in{\mathcal {M}}_{+}(S^{n-1},\hat{e})\) such that
where \({\mathcal {M}}_{+}(S^{n-1},\hat{e})\) denotes the set of nonnegative zonal measures on \(S^{n-1}\).
For the mixed radial Blaschke-Minkowski homomorphism induced by Ψ, Schuster [5] proved that
Obviously, a special case is the following:
where i are integers. We now extend the integers i to real numbers, define the Blaschke-Minkowski homomorphism of order p of K.
Definition 4.4
Let \(K\in{\mathcal {S}}^{n}\), the Blaschke-Minkowski homomorphism of order p of K, denoted by \(\Psi_{p}K\), is defined for all \(p\in{\Bbb {R}}\) by
This extended definition will be required to prove our main results.
5 Inequalities for the radial Blaschke-Minkowski homomorphism
Theorem 5.1
Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq p< n-1\) and \(i\leq n-1\leq j\leq n\), then
with equality if and only if \(\Psi_{p} K\) and \(\Psi_{p} L\) are dilates.
Remark 5.2
Taking \(j=n\) in (5.1) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (5.1) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(i\leq n-1\), then
with equality if and only if \(\Psi_{p} K\) and \(\Psi_{p} L\) are dilates. Taking \(p=0\) in (5.1), (5.1) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\) and \(i\leq n-1\leq j\leq n\), then
with equality if and only if ΨK and ΨL are dilates.
Theorem 5.3
Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq i< n \), \(p< i-1\) and \(k,j\in{\Bbb {R}}\) satisfy \(j\leq n-1\leq k\leq n\), then
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.
Remark 5.4
Taking \(k=n\) in (5.4) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (5.4) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\), \(p< i-1\) and \(j\leq n-1\), then
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates. Taking \(i=0\), \(j=0\) and \(k=n\) in (5.4), we have: If \(K,L\in{\mathcal {S}}^{n}\) and \(p<-1\), then
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.
6 Dresher’s inequalities for p-radial Blaschke and harmonic Blaschke additions
An extension of Beckenbach’s inequality (see [3], p. 27) was obtained by Dresher [25] by means of moment-space techniques.
Lemma 6.1
(Dresher’s inequality)
If \(p\geq1\geq r\geq0\), \(f,g\geq0\) and Ï• is a distribution function, then
with equality if and only if the functions f and g are proportional.
We are now in a position to prove Theorem 5.1. The following statement is just a slight reformulation of it.
Theorem 6.2
Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq p< n-1\) and \(s,t\in{\Bbb {R}}\) satisfy \(s\geq1\geq t\geq0\), then
with equality if and only if \(\Psi_{p} K\) and \(\Psi_{p} L\) are dilates.
Proof
From (2.4), we obtain
where μ is the generating measure of Ψ from Lemma 4.3. Hence, from (4.6), we obtain
Therefore, by (1.6), we have
and
From (6.3), (6.4) and Lemma 6.1, we obtain
Equality holds if and only if the functions \(\rho(\Psi_{p}K,u)\) and \(\rho(\Psi_{p}L,u)\) are proportional.
Taking \(s=n-i\) and \(t=n-j\) in Theorem 6.2, Theorem 6.2 becomes Theorem 5.1 stated in Section 5. If \(\Psi:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow {\mathcal {S}}^{n}\) is the mixed intersection operator \({\bf I}:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow{\mathcal {S}}^{n}\) in (6.2) and \(n-s=i\) and \(n-t=j\), we obtain the following result: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(i\leq n-1\leq j\leq n\), then
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. Taking \(j=n\) in (6.5) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (6.5) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(i\leq n-1\), then
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. □
We are now in a position to prove Theorem 5.3. The following statement is just a slight reformulation of it.
Theorem 6.3
Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq i< n \), \(p< i-1\) and \(s,t\in{\Bbb {R}}\) satisfy \(s\geq1\geq t\geq0\), then
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.
Proof
From (3.3), we obtain
Hence, from (4.6), we obtain
By (1.6), we have
and
From (6.7), (6.8) and Lemma 6.1, we obtain
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.
Taking \(s=n-j\) and \(t=n-k\) in Theorem 6.3, Theorem 6.3 becomes Theorem 5.3 stated in Section 5. If \(\Psi:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow {\mathcal {S}}^{n}\) is the mixed intersection operator \({\bf I}:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow{\mathcal {S}}^{n}\) in (6.6) and \(j=n-s\) and \(k=n-t\), we obtain the following result: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\), \(p< i-1\) and \(j\leq n-1\leq k\leq n\), then
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. Taking \(k=n\) in (6.9) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (6.9) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\), \(p< i-1\) and \(j\leq n-1\), then
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. □
7 Conclusions
In the present study, we first revised and improved the concepts of radial Blaschke addition and harmonic Blaschke addition in an \(L_{p}\) space. Following this, we established Dresher’s inequalities (Brunn-Minkowski type) for the radial Blaschke-Minkowski homomorphisms with respect to the p-radial addition and the p-harmonic Blaschke addition.
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The author’s research is supported by the Natural Science Foundation of China (11371334).
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Zhao, CJ. On p-radial Blaschke and harmonic Blaschke additions. J Inequal Appl 2017, 308 (2017). https://doi.org/10.1186/s13660-017-1581-y
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DOI: https://doi.org/10.1186/s13660-017-1581-y