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On dual mixed quermassintegral quotient functions
Journal of Inequalities and Applications volume 2015, Article number: 340 (2015)
Abstract
We introduce the notion of dual mixed quermassintegral quotient functions and establish the Brunn-Minkowski inequalities for them in this paper.
1 Introduction and main results
The setting for this paper is Euclidean n-space \(\mathbb{R}^{n}\). Let \(\mathcal{S}^{n}_{o}\) denote the set of star bodies containing the origin in their interiors in \(\mathbb{R}^{n}\). Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\), and let \(V(K)\) denote the n-dimensional volume of body K. For the standard unit ball B in \(\mathbb{R}^{n}\), we use \(\omega_{n}=V(B)\) to denote its volume.
In 1975, Lutwak (see [1]) gave the notion of dual mixed volumes as follows: For \(K_{1}, K_{2}, \ldots, K_{n}\in\mathcal{S}^{n}_{o}\), the dual mixed volume, \(\widetilde{V}(K_{1}, K_{2}, \ldots, K_{n})\), of \(K_{1}, K_{2}, \ldots, K_{n}\) is defined by
Taking \(K_{1}=\cdots=K_{n-i}=K\), \(K_{n-i+1}=\cdots=K_{n}=L\) in (1.1), we write \(\widetilde{V}_{i}(K, L)=\widetilde{V}(K, n-i; L, i )\), where K appears \(n-i\) times and L appears i times. Then
Let \(L=B\) in (1.2) and notice \(\rho(B, \cdot)=1\), and allow i is any real, then the dual quermassintegrals can be defined as follows: For \(K\in\mathcal{S}^{n}_{o}\) and i is any real, the dual quermassintegrals, \(\widetilde{W}_{i}(K)\), of K are given by (see [1])
Associated with dual quermassintegrals, Zhao (see [2]) defined the dual quermassintegral quotient functions of a star body K by
Further, in [2] the Brunn-Minkowski type inequalities for the dual quermassintegral quotient functions of star bodies were established as follows.
Theorem A
If \(K, L\in\mathcal{S}^{n}_{o}\) and reals i, j satisfy \(i\leq n-1\leq j\leq n\), then
Here \(\tilde{+}\) is the radial Minkowski sum.
Theorem B
If \(K, L\in\mathcal{S}^{n}_{o}\) and reals i, j satisfy \(i\leq1\leq j\leq n\), then
Here \(\breve{+}\) is the radial Blaschke sum.
Theorem C
If \(K, L\in\mathcal{S}^{n}_{o}\) and reals i, j satisfy \(i\leq-1\leq j\leq n\), then
Here \(\hat{+}\) is the harmonic Blaschke sum.
Motivated by the work of Zhao, we give the following definition of dual mixed quermassintegral quotient function.
Let \(K_{1}=\cdots=K_{n-i-1}=K\), \(K_{n-i}=\cdots=K_{n-1}=B\), \(K_{n}=L\) in (1.1), then we write \(\widetilde{W}_{i}(K, L)=\widetilde{V}(K, n-i-1; B, i; L, 1)\), where K appears \(n-i-1\) times, B appears i times and L appears 1 time. Here, we allow i to be any real and define as follows: For \(K, L\in\mathcal{S}^{n}_{o}\) and i any real, the dual mixed quermassintegrals, \(\widetilde{W}_{i}(K, L)\), of K and L are given by
Obviously, from (1.3) and (1.5), we have \(\widetilde{W}_{i}(K, K)=\widetilde{W}_{i}(K)\). According to (1.5), we define the following.
Definition 1.1
Let \(K, L\in\mathcal{S}^{n}_{o}\) and \(i,j\in\mathbb{R}\), the dual mixed quermassintegral quotient function, \(Q_{\widetilde{W}_{i,j}(K,L)}\), of K and L can be defined by
Obviously, if \(L=K\), then (1.6) is just (1.4).
The aim of this paper is to establish the following Brunn-Minkowski type inequalities for dual mixed quermassintegral quotient functions of star bodies.
Theorem 1.1
For \(K, K', L\in\mathcal {S}^{n}_{o}\), if \(i\leq n-2\leq j< n-1\), then
if \(n-2\leq i< n-1<j\), then
In each case, equality holds if and only if K and \(K'\) are dilates. Here \(\tilde{+}\) is the radial Minkowski sum.
Theorem 1.2
For \(K, K', L\in\mathcal {S}^{n}_{o}\), if \(i\leq0\leq j< n-1\), then
if \(0\leq i< n-1< j\), then
In each case, equality holds if and only if K and \(K'\) are dilates. Here \(\breve{+}\) is the radial Blaschke sum.
Theorem 1.3
For \(K, K', L\in\mathcal {S}^{n}_{o}\), if \(i\leq-2\leq j< n-1\), then
if \(-2\leq i< n-1< j\), then
In each case, equality holds if and only if K and \(K'\) are dilates. Here \(\hat{+}\) is the harmonic Blaschke sum.
2 Preliminaries
For a compact set K in \(\mathbb{R}^{n}\) which is star shaped with respect to the origin, we define the radial function \(\rho_{K}(u)=\rho(K,u)\) of K by
If \(\rho_{K}\) is positive and continuous, K will be called a star body (about the origin). Two star bodies K and L are said to be dilates (of one another) if \(\rho_{K}(u)/\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
For \(K_{1}, K_{2}\in\mathcal{S}^{n}_{o}\), and \(\lambda_{1},\lambda_{2}\geq 0\) (not both 0), the radial function of the radial Minkowski linear combination \(\lambda_{1}K_{1}\,\tilde{+}\,\lambda_{2}K_{2}\) is given by Zhang (see [3]):
For \(K_{1}, K_{2}\in\mathcal{S}^{n}_{o}\), and \(\lambda_{1},\lambda_{2}\geq 0\) (not both 0), the radial Blaschke linear combination \(\lambda _{1}\cdot K_{1} \,\check{+}\,\lambda_{2}\cdot K_{2}\) is a star body whose radial function is given by Lutwak (see [4]):
For \(K_{1}, K_{2}\in\mathcal{S}^{n}_{o}\), and \(\lambda_{1},\lambda_{2}\geq 0\) (not both 0), the harmonic Blaschke linear combination \(\lambda _{1} \circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2}\) is a star body whose radial function is given by Lutwak (see [5]):
3 Proofs of theorems
According to a generalization of the Dresher inequality (see [6]), we get the reverse Dresher inequality.
Lemma 3.1
(Dresher’s inequality)
Let functions \(f_{1}, f_{2}, g_{1}, g_{2} \geq0\), E is a bounded measurable subset in \(\mathbb{R}^{n}\). If \(p \geq1 \geq r \geq0\), then
equality holds if and only if \(f_{1}/ f_{2}= g_{1}/g_{2} \).
Lemma 3.2
(Reverse Dresher’s inequality)
Let functions \(f_{1}, f_{2}, g_{1}, g_{2} \geq0\), E is a bounded measurable subset in \(\mathbb{R}^{n}\). If \(1\geq p>0>r\), then
equality holds if and only if \(f_{1}/ f_{2}= g_{1}/g_{2} \).
Proof of Lemma 3.2
If \(f_{1}, f_{2}, g_{1}, g_{2} \geq0\), and \(1\geq p>0>r\), according to the Minkowski inequality,
For \(1\geq p>0>r\), we have
According to the Hölder inequality, \(\frac{p-r}{p}>1\), and (3.3), (3.4),
According to the equality condition of the Minkowski inequality and the Hölder inequality, equality holds in (3.2) if and only if \(f_{1}/ f_{2}= g_{1}/g_{2} \). □
Proof of Theorem 1.1
From (2.1), for \(K, K', L\in\mathcal {S}^{n}_{o}\),
and
From (3.1), (3.5), and (3.6), for \(p\geq1\geq r>0\), we have
According to the equality condition of inequality (3.1), we see that equality holds in (3.7) if and only if K and L, \(K'\), and L are dilates, respectively. So K and \(K'\) are dilates.
Let \(i=n-p-1\), \(j=n-r-1\), then \(p\geq1\geq r>0\) and \(i\leq n-2\leq j< n-1\) are equivalent. This and (3.7) yield inequality (1.7) and its equality condition.
Similarly, if \(1\geq p>0>r\), according to (3.2), (3.5), and (3.6), we have
and equality holds if and only if K and \(K'\) are dilates.
Let \(i=n-p-1\), \(j=n-r-1\), then (3.8) gives inequality (1.8) and its equality condition. □
Proof of Theorem 1.2
From (2.2), for \(K, K', L \in\mathcal {S}^{n}_{o}\), we have
and
According to (3.1), (3.9), and (3.10), for \(p\geq n-1\geq r>0\),
Then
According to the equality condition of inequality (3.1), we see that equality holds in (3.11) if and only if K and \(K'\) are dilates.
Let \(i=n-p-1\) and \(j=n-r-1\), then \(p\geq n-1\geq r>0\) and \(i\leq0\leq j< n-1\) are equivalent. This and (3.11) yield inequality (1.9) and its equality condition.
Similarly, if \(n-1\geq p>0>r\), according to (3.2), (3.9), and (3.10), we have
and equality holds if and only if K and \(K'\) are dilates.
Let \(i=n-p-1\) and \(j=n-r-1\), then (3.12) gives inequality (1.10) and its equality condition. □
Proof of Theorem 1.3
From (2.3), for \(K, K', L \in\mathcal {S}^{n}_{o}\),
and
According to (3.1), (3.13), and (3.14), for \(p\geq n+1>r>0\),
i.e.,
According to the equality condition of inequality (3.1), we see that equality holds in (3.15) if and only if K and \(K'\) are dilates.
Let \(i=n-p-1\) and \(j=n-r-1\), then \(p\geq n+1\geq r>0\) and \(i\leq-2\leq j< n-1\) are equivalent. This and (3.15) yield inequality (1.11) and its equality condition.
If \(n+1\geq p>0>r\), according to (3.2), (3.13), and (3.14), we have
with equality if and only if K and \(K'\) are dilates.
Let \(i=n-p-1\) and \(j=n-r-1\), then (3.16) gives inequality (1.12) and its equality condition. □
References
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Acknowledgements
This work was supported by the Natural Science Foundation of China (Grant Nos. 11371224 and 11102101).
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The main idea of this paper was proposed by the second author. All authors contributed equally to the writing of the paper. All authors read and approved the final manuscript.
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Zhang, P., Wang, W. & Zhang, X. On dual mixed quermassintegral quotient functions. J Inequal Appl 2015, 340 (2015). https://doi.org/10.1186/s13660-015-0871-5
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DOI: https://doi.org/10.1186/s13660-015-0871-5