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General \(L_{p}\)-mixed-brightness integrals
Journal of Inequalities and Applications volume 2015, Article number: 190 (2015)
Abstract
The notion of mixed-brightness integrals was introduced by Li and Zhu. In this paper, motivated by the notion of general \(L_{p}\)-projection bodies, introduced by Haberl and Schuster, we define general \(L_{p}\)-mixed-brightness integrals and determine their extremal values, as well as several other inequalities for them.
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 containing the origin in their interiors and the set of origin-symmetric convex bodies in \(\mathbb {R}^{n}\), we write \(\mathcal{ K}^{n}_{o}\) and \(\mathcal{ K}^{n}_{os}\), respectively. Let \(\mathcal{ S}^{n}_{o}\) denote the set of star bodies (about the origin) in \(\mathbb{R}^{n}\) and let \(S^{n-1}\) denote the unit sphere in \(\mathbb {R}^{n}\). By \(V(K)\) we denote the n-dimensional volume of a body K and for the standard unit ball B in \(\mathbb{R}^{n}\), we write \(\omega _{n}\) for its volume.
If \(K\in\mathcal{ K}^{n}\), then its support function, \(h_{K}=h(K, \cdot ):\mathbb{R}^{n}\rightarrow(-\infty, \infty)\), is defined by [1, 2]
where \(x\cdot y\) denotes the standard inner product of x and y.
Projection bodies of convex bodies were introduced at the turn of the previous century by Minkowski [1]. For \(K\in\mathcal{ K}^{n}\), the projection body, Î K, of K is the origin-symmetric convex body, defined by
for all \(u\in S^{n-1}\). Here \(S(K, \cdot)\) denotes the surface area measure of K.
Using the classical notion of projection bodies, Li and Zhu [3] recently introduced the mixed-brightness integral: For \(K_{1}, \ldots, K_{n}\in\mathcal{ K}^{n}\), the mixed-brightness integral, \(D(K_{1}, \ldots, K_{n})\), is defined by
where \(\delta(K, u)=\frac{1}{2}h(\Pi K, u)\) is the half brightness of \(K\in\mathcal{ K}^{n}\) in the direction u. Convex bodies \(K_{1},\ldots,K_{n}\) are said to have similar brightness if there exist constants \(\lambda _{1},\ldots,\lambda_{n}>0\) such that \(\lambda_{1}\delta(K_{1},u)=\lambda _{2}\delta(K_{2},u)=\cdots=\lambda_{n}\delta(K_{n},u)\) for all \(u\in S^{n-1}\).
Further, Li and Zhu [3] established the following Fenchel-Aleksandrov type inequality for mixed-brightness integrals.
Theorem 1.A
If \(K_{1},\ldots,K_{n} \in\mathcal{ K}^{n}\) and \(1< m\leq n\), then
with equality if and only if \(K_{n-m+1},K_{n-m+2},\ldots,K_{n}\) are all of similar brightness.
More recently, Zhou et al. [4] obtained Brunn-Minkowski type inequalities for mixed-brightness integrals.
The notion of \(L_{p}\)-projection bodies was introduced by Lutwak et al. [5]. For each \(K \in\mathcal{ K}^{n}_{o}\) and \(p\geq1\), the \(L_{p}\)-projection body, \(\Pi_{p} K\), is the origin-symmetric convex body whose support function is defined by
for all \(u\in S^{n-1}\), where \(\alpha_{n,p}=1/n\omega_{n}c_{n-2,p}\) with \(c_{n,p}=\omega_{n+p}/\omega_{2}\omega_{n}\omega_{p-1}\), and \(S_{p}(K, \cdot)\) is the \(L_{p}\)-surface measure of K. The normalization in definition (1.3) is chosen such that \(\Pi_{p}B = B\).
As part of the tremendous progress in the theory of Minkowski valuations (see [6–14]), Ludwig [15] discovered more general \(L_{p}\)-projection bodies \(\Pi_{p}^{\tau}K\in\mathcal{ K}^{n}_{o}\), which can be defined using the function \(\varphi _{\tau}:\mathbb{R}\rightarrow[0,\infty)\) given by
where \(\tau\in[-1,1]\). Now for \(K\in\mathcal{ K}^{n}_{o}\) and \(p\geq1\), let \(\Pi_{p}^{\tau}K\in\mathcal{ K}^{n}_{o}\) with support function
where
The normalization is again chosen such that \(\Pi^{\tau}_{p}B=B\) for every \(\tau\in[-1,1]\). Obviously, if \(\tau=0\), then \(\Pi^{\tau}_{p}K=\Pi_{p}K\).
For general \(L_{p}\)-projection bodies, Haberl and Schuster [16] proved the general \(L_{p}\)-Petty projection inequality and determined the extremal values of volume for polars of general \(L_{p}\)-projection bodies. Wang and Wan [17] investigated Shephard type problems for general \(L_{p}\)-projection bodies. Wang and Feng [18] established general \(L_{p}\)-Petty affine projection inequality. These investigations were the starting point of a new and rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [13–32]).
In this article, using the notion of general \(L_{p}\)-projection bodies, we define general \(L_{p}\)-mixed-brightness integrals as follows: For \(K_{1}, \ldots, K_{n}\in\mathcal{ K}_{o}^{n}\), \(p\geq1\) and \(\tau\in[-1,1]\), the general \(L_{p}\)-mixed-brightness integral, \(D^{(\tau)}_{p}(K_{1}, \ldots , K_{n})\), of \(K_{1}, \ldots, K_{n}\) is defined by
where \(\delta^{(\tau)}_{p}(K, u)=\frac{1}{2}h(\Pi^{\tau}_{p} K, u)\) denotes the half general \(L_{p}\)-brightness of \(K\in\mathcal{ K}_{o}^{n}\) in the direction u. Convex bodies \(K_{1},\ldots,K_{n}\) are said to have similar general \(L_{p}\)-brightness if there exist constants \(\lambda_{1},\ldots ,\lambda_{n}>0\) such that, for all \(u\in S^{n-1}\),
Remark 1.1
For \(\tau=0\) in (1.5), we write \(D^{(\tau)}_{p}(K_{1}, \ldots, K_{n})=D_{p}(K_{1}, \ldots, K_{n})\) and \(\delta^{(\tau)}_{p}(K,u)=\delta_{p}(K,u)\) for all \(u\in S^{n-1}\). Then
where \(\delta_{p}(K, u)=\frac{1}{2}h(\Pi_{p} K, u)\). Here \(D_{p}(K_{1}, \ldots, K_{n})\) is called the \(L_{p}\)-mixed-brightness integral of \(K_{1}, \ldots, K_{n}\in\mathcal{ K}_{o}^{n}\). Obviously, for \(p=1\), (1.6) is just the mixed-brightness integral from (1.1).
Let \(\underbrace{K_{1}=\cdots=K_{n-i}}_{n-i}=K\) and \(\underbrace {K_{n-i+1}=\cdots=K_{n}}_{i}=L\) (\(i=0, 1, \ldots, n\)) in (1.5), we denote \(D^{\tau}_{p,i}(K,L)=D^{(\tau)}_{p}(\underbrace{K, \ldots, K}_{n-i}, \underbrace{L, \ldots, L}_{i})\). More general, if i is any real, we define for \(K, L\in\mathcal{ K}_{o}^{n}\), \(p\geq1\), and \(\tau\in [-1,1]\), the general \(L_{p}\)-mixed-brightness integral, \(D^{\tau}_{p,i}(K, L)\), of K and L by
For \(L=B\) in (1.7), we write \(D^{(\tau)}_{p,i}(K, B)=\frac {1}{2^{i}}D^{(\tau)}_{p,i}(K)\) and notice that \(\delta^{(\tau)}_{p} (B, u)=\frac{1}{2}h(\Pi^{\tau}_{p} B, u)=\frac{1}{2}\) for all \(u\in S^{n-1}\), which together with (1.7) yields
where \(D^{(\tau)}_{p,i}(K)\) is called the ith general \(L_{p}\)-mixed-brightness integral of K. If \(\tau=0\), then \(D^{(\tau )}_{p,i}(K)=D_{p,i}(K)\). For \(\tau= \pm1\), we write \(D^{(\tau)}_{p,i}(K)=D^{\pm}_{p,i}(K)\).
For \(L=K\) in (1.7), write \(D^{(\tau)}_{p,i}(K, K)=D^{(\tau)}_{p}(K)\), which is called the general \(L_{p}\)-brightness integral of K. Clearly,
Obviously, by (1.5), (1.7), (1.8), and (1.9), we have
In this paper, we establish several inequalities for general \(L_{p}\)-mixed-brightness integrals. First, we determine the extremal values of general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.1
If \(K\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), then
If K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality if and only if \(\tau= 0\) and equality in the right inequality if and only if \(\tau= \pm1\).
Next, we obtain a Brunn-Minkowski type inequality for general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.2
If \(K, L\in\mathcal{ K}^{n}_{os}\), \(p\geq1\), \(\tau\in[-1,1]\), and \(i\in \mathbb{R}\), and such that \(i\neq n\), then for \(i< n-p\),
For \(n-p< i< n\) or \(i>n\), we have
In each case, equality holds if and only if K and L have similar general \(L_{p}\)-brightness. For \(i=n-p\), equality always holds in (1.13) or (1.14).
Here, \(\lambda\circ K\oplus_{p} \mu\circ L\) denotes the \(L_{p}\)-Blaschke combination of K and L.
Next, we extend inequality (1.2) to general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.3
If \(K_{1}, \ldots, K_{n}\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), \(\tau\in[-1,1]\), and \(1< m\leq n\), then
with equality if and only if \(K_{n-m+1},K_{n-m+2},\ldots,K_{n}\) are all of similar general \(L_{p}\)-brightness.
Taking \(m=n\) in Theorem 1.3 and using (1.10), we obtain the following corollary.
Corollary 1.1
If \(K_{1}, \ldots, K_{n}\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in [-1,1]\), then
with equality if and only if \(K_{1},K_{2},\ldots,K_{n}\) are all of similar general \(L_{p}\)-brightness.
Moreover, we also establish the following cyclic inequality for general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.4
If \(K,L\in\mathcal {K}^{n}_{o}\), \(p\geq1\), \(\tau\in[-1,1]\), and \(i,j,k\in\mathbb{R}\) such that \(i< j< k\), then
with equality if and only if K and L have similar general \(L_{p}\)-brightness.
Taking \(i=0\), \(k=n\) in Theorem 1.4 and using (1.11), we obtain the following result.
Corollary 1.2
If \(K,L\in\mathcal{K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), then for \(0< j< n\),
with equality if and only if K and L have similar general \(L_{p}\)-brightness. For \(j=0\) or \(j=n\), equality always holds in (1.17).
Let \(L=B\) in Theorem 1.4, we also have the following result.
Corollary 1.3
If \(K\in\mathcal{K}^{n}_{o}\), \(p\geq 1\), \(\tau\in[-1,1]\), and \(i,j,k\in\mathbb{R}\) such that \(i< j< k\), then
with equality if and only if K and L have similar general \(L_{p}\)-brightness, i.e., K has constant general \(L_{p}\)-brightness.
2 Notation and background material
2.1 Radial function and polars of convex bodies
If K is a compact star-shaped set (about the origin) in \(\mathbb{R}^{n}\), then its radial function, \(\rho_{K}=\rho(K,\cdot):\mathbb{R}^{n}\rightarrow[0,\infty)\), is defined by (see [1])
If \(\rho_{K}\) is positive and continuous, K will be called a star body (with respect to 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 \(u\in S^{n-1}\).
If E is a nonempty set in \(\mathbb{R}^{n}\), then the polar set of E, \(E^{\ast}\), is defined by (see [1])
From this, we see that (see [1]) if \(K\in \mathcal{ K}^{n}_{o}\), then \((K^{\ast})^{\ast}=K\) and
Lutwak in [33] defined dual quermassintegrals as follows. For \(K\in S^{n}_{o}\) and any real i, the dual quermassintegral, \(\widetilde{ W}_{i}(K)\), of K is defined by
Obviously, (2.3) implies that
2.2 \(L_{p}\)-combinations of convex and star bodies
For \(K, L\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the Firey \(L_{p}\)-combination, \(\lambda\cdot K+_{p}\mu\cdot L\in\mathcal{ K}^{n}_{o}\), of K and L is defined by (see [34, 35])
where the symbol â‹… in \(\lambda\cdot K\) denotes the Firey scalar multiplication. Note that \(\lambda\cdot K=\lambda^{1/p}K\).
For \(K, L\in\mathcal{S}^{n}_{o}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-harmonic radial combination, \(\lambda\star K+_{-p}\mu\star L\in\mathcal{S}^{n}_{o}\), of K and L is defined by (see [36])
where \(\lambda\star K=\lambda^{-1/p}K\).
From (2.2), (2.5), and (2.6), we easily find that if \(K, L\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), then
In [37] Wang and Leng established the following Brunn-Minkowski type inequality for dual quermassintegrals with respect to an \(L_{p}\)-harmonic radial combination of star bodies.
Theorem 2.A
If \(K,L\in\mathcal{S}^{n}_{o}\), \(p\geq1\), \(i\in\mathbb{R}\) and such that \(i\neq n\), and \(\lambda, \mu\geq0\) (not both zero), then for \(i< n\) or \(n< i< n+p\),
for \(i>n+p\),
In each inequality, equality holds if and only if K and L are dilates. For \(i=n+p\), equality always holds in (2.8) and (2.9).
The \(L_{p}\)-Blaschke combination of origin-symmetric convex bodies was introduced by Lutwak [35]. For \(K, L\in\mathcal{ K}^{n}_{os}\), \(p\geq 1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-Blaschke combination, \(\lambda\circ K\oplus_{p}\mu\circ L\in \mathcal{ K}^{n}_{os}\), of K and L is defined by
where \(\lambda\circ K=\lambda^{1/(n-p)}K\). For more information on these and other binary operations between convex and star bodies, see [38–42].
2.3 General \(L_{p}\)-projection bodies
For \(p\geq1\), Ludwig [15] discovered the asymmetric \(L_{p}\)-projection body, \(\Pi^{+}_{p}K\), of \(K\in\mathcal{ K}^{n}_{o}\), whose support function is defined by
where \((u\cdot v)_{+}=\max\{u\cdot v, 0\}\). In [16], Haberl and Schuster also defined
Using definition (1.4) of general \(L_{p}\)-projection bodies, Haberl and Schuster [16] showed that, for \(K\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\),
where
Moreover, they [16] determined the following extremal values of the volume for polars of general \(L_{p}\)-projection bodies.
Theorem 2.B
If \(K\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
If K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\).
Here, \(\Pi^{\tau,\ast}_{p}K\) denotes the polar of the general \(L_{p}\)-projection body \(\Pi^{\tau}_{p}K\).
3 Proofs of the main theorems
In this section, we will prove Theorems 1.1-1.3.
To complete the proofs of Theorems 1.1-1.2, we require the following a lemma.
Lemma 3.1
If \(K\in\mathcal{ K}_{o}^{n}\), \(p\geq1\), \(\tau\in[-1,1]\), and i is any real, then
Proof
By (1.8), (2.2), and (2.3), we have
 □
Proof of Theorem 1.1
Taking \(i=2n\) in (3.1) and using (2.4), we obtain
Therefore, by inequality (2.11) together with (3.2), we immediately obtain
This is inequality (1.12).
According to the equality conditions of inequality (2.11), we know that if K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality of (1.12) if and only if \(\tau= 0\) and equality in the right inequality of (1.12) if and only if \(\tau= \pm 1\). □
Proof of Theorem 1.2
By (1.4) and (2.10), we have, for all \(u\in S^{n-1}\),
i.e.,
This together with (2.7), yields
Hence, if \(i< n-p\), then \(2n-i>n+p\). From this, (3.1), (3.3), and inequality (2.9), we obtain
This yields inequality (1.13).
From the equality conditions of inequality (2.9), we see that equality holds in (1.13) if and only if \(\Pi_{p}^{\tau,\ast}K\) and \(\Pi _{p}^{\tau,\ast}L\) are dilates, i.e., \(\Pi_{p}^{\tau}K\) and \(\Pi_{p}^{\tau}L\) are dilates. This means equality holds in (1.13) if and only if K and L have similar general \(L_{p}\)-brightness.
Similarly, if \(n-p< i< n\) or \(i>n\), then \(2n-i< n\) or \(n<2n-i<n+p\). Thus, using (3.1), (3.3), and inequality (2.8), we obtain inequality (1.14).
If \(i=n-p\), then \(2n-i=n+p\). This combined with Theorem 2.A, shows that equality always holds in (1.13) or (1.14). □
The proof of Theorem 1.3 requires the following inequality [3].
Lemma 3.2
If \(f_{0},f_{1},\ldots, f_{m}\) are (strictly) positive continuous functions defined on \(S^{n-1}\) and \(\lambda_{1},\ldots,\lambda_{m}\) are positive constants the sum of whose reciprocals is unity, then
with equality if and only if there exist positive constants \(\alpha _{1},\alpha_{2},\ldots,\alpha_{m}\) such that \(\alpha_{1}f_{1}^{\lambda _{1}}(u)=\cdots=\alpha_{m}f_{m}^{\lambda_{m}}(u)\) for all \(u\in S^{n-1}\).
Proof of Theorem 1.3
For \(K_{1},\ldots,K_{n}\in\mathcal{ K}^{n}_{o}\), take \(\lambda_{i}=m\) in (3.4) (\(1\leq i\leq n\)), and
Then we have
i.e.
According to the equality conditions of Lemma 3.2, we see that equality holds in (3.5) if and only if there exist positive constants \(\lambda _{1},\lambda_{2},\ldots,\lambda_{m}\) such that
for all \(u\in S^{n-1}\). Thus equality holds in (1.15) if and only if \(K_{n-m+1},K_{n-m+2},\ldots,K_{n}\) are all of similar general \(L_{p}\)-brightness. □
Proof of Theorem 1.4
From (1.7) and the Hölder inequality, we obtain
This gives the desired inequality (1.16). According to the equality conditions of the Hölder inequality, we know that equality holds in (1.16) if and only if there exists a constant \(\lambda>0\) such that
i.e. \(\delta^{(\tau)}_{p}(K,u)=\lambda\delta^{(\tau )}_{p}(L,u)\) for all \(u\in S^{n-1}\). Thus equality holds in (1.16) if and only if K and L have similar general \(L_{p}\)-brightness. □
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Acknowledgements
The authors would like to sincerely thank the referees for all valuable and helpful comments and suggestions which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No. 2015PV070).
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Yan, L., Wang, W. General \(L_{p}\)-mixed-brightness integrals. J Inequal Appl 2015, 190 (2015). https://doi.org/10.1186/s13660-015-0708-2
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DOI: https://doi.org/10.1186/s13660-015-0708-2