Lutwak defined the dual Blaschke combination of star bodies. In this paper, based on the \(L_{p}\)-dual Blaschke combination of star bodies, we define the general \(L_{p}\)-dual Blaschke bodies and obtain the extremal values of their volume and \(L_{p}\)-dual affine surface area. Further, as the applications, we study two negative forms of the \(L_{p}\)-Busemann-Petty problems.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
1 Introduction and main results
Let \({\mathcal{K}}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space \(\mathbb{R}^{n}\). \({\mathcal{K}}^{n}_{c}\) denotes the set of convex bodies whose centroid lies at the origin in \(\mathbb{R}^{n}\). Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\) and \(V(K)\) denote the n-dimensional volume of a body K. For the standard unit ball B in \(\mathbb{R}^{n}\), its volume is written by \(\omega_{n} = V(B)\).
If K is a compact star shaped (about the origin) in \(\mathbb{R}^{n}\), then its radial function \(\rho_{K}=\rho(K,\cdot)\) is defined on \(S^{n-1}\) by letting (see [1, 2])
If \(\rho_{K}\) is positive and continuous, then K will be called a star body (about the origin). For the set of star bodies containing the origin in their interiors and the set of origin-symmetric star bodies in \(\mathbb{R}^{n}\), we write \({\mathcal{S}}^{n}_{o}\) and \({\mathcal{S}}^{n}_{os}\), respectively. Two star bodies K and L are said to be dilates (of one another) if \(\rho_{K}(u)/\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
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The notion of dual Blaschke combination was given by Lutwak (see [3]). For \(K, L\in{\mathcal{S}}_{o}^{n}\), \({\lambda, \mu\geq 0}\) (not both zero), \(n \geq2\), the dual Blaschke combination \(\lambda\circ K \oplus\mu\circ L\in{\mathcal{S}}_{o}^{n}\) of K and L is defined by
$$\rho(\lambda\circ K \oplus\mu\circ L,\cdot)^{n-1}= \lambda\rho(K, \cdot)^{n-1}+\mu\rho(L,\cdot)^{n-1}, $$
where the operation ‘⊕’ is called dual Blaschke addition and \(\lambda\circ K\) denotes dual Blaschke scalar multiplication.
Combining with the definition of dual Blaschke combination, Lutwak [3] gave the concept of dual Blaschke body as follows: For \(K \in \mathcal{S}_{o}^{n}\), take \(\lambda= \mu=1/2\), \(L=-K\), the dual Blaschke body \(\overline{\nabla}K\) is given by
$$\overline{\nabla}K=\frac{1}{2} \circ K \oplus\frac {1}{2}\circ(-K). $$
In this paper, we define the notion of \(L_{p}\)-dual Blaschke combination as follows: For \(K, L\in{\mathcal{S}}_{o}^{n}\), \({\lambda, \mu \geq0}\) (not both zero), \(n>p>0\), the \(L_{p}\)-dual Blaschke combination \(\lambda\circ K \oplus_{p} \mu\circ L\in{\mathcal{S}}_{o}^{n}\) of K and L is defined by
$$ \rho(\lambda\circ K \oplus_{p} \mu\circ L,\cdot)^{n-p}= \lambda\rho(K,\cdot)^{n-p}+\mu\rho(L,\cdot)^{n-p}, $$
(1.1)
where the operation ‘\(\oplus_{p}\)’ is called \(L_{p}\)-dual Blaschke addition and \(\lambda\circ K=\lambda^{\frac{1}{n-p}}K\).
Let \(\lambda=\mu=\frac{1}{2}\) and \(L=-K\) in (1.1), then the \(L_{p}\)-dual Blaschke body \(\overline{\nabla}_{p}K\) of \(K\in {\mathcal{S}}_{o}^{n}\) is given by
Now, by (1.1) we define the general \(L_{p}\)-dual Blaschke bodies as follows: For \(K\in{\mathcal{S}}_{o}^{n}\), \(n > p >0\) and \(\tau\in[-1, 1]\), the general \(L_{p}\)-dual Blaschke body \(\overline{\nabla}_{p}^{\tau}K\) of K is defined by
$$ \overline{\nabla}_{p}^{\tau}K=f_{1}(\tau)\circ K \oplus_{p} f_{2}(\tau )\circ(-K). $$
(1.7)
Besides, by (1.2), (1.4) and (1.7), we see that if \(\tau=0\), then \(\overline{\nabla}_{p}^{0} K=\overline{\nabla}_{p} K\); if \(\tau=\pm1\), then \(\overline{\nabla}_{p}^{+1} K=K\), \(\overline{\nabla}_{p}^{-1} K=-K\).
The main results of this paper can be stated as follows: First, we give the extremal values of the volume of general \(L_{p}\)-dual Blaschke bodies.
Theorem 1.1
If\(K \in{\mathcal{S}}_{o}^{n}\), \(n>p> 0\), \(\tau\in[-1, 1]\), then
If\(\tau\neq0\), equality holds in the left inequality if and only ifKis origin-symmetric, if\(\tau\neq\pm 1\), then equality holds in the right inequality if and only ifKis also origin-symmetric.
Moreover, based on the \(L_{p}\)-dual affine surface area \(\widetilde {\Omega}_{p}(K)\) of \(K \in{\mathcal{S}}_{o}^{n}\) (see (2.7)), we give another class of extremal values for general \(L_{p}\)-dual Blaschke bodies.
Theorem 1.2
If\(K \in{\mathcal{S}}_{o}^{n}\), \(n>p> 0\), \(\tau\in[-1, 1]\), then
If\(\tau\neq0\), equality holds in the left inequality if and only ifKis origin-symmetric, if\(\tau\neq\pm 1\), then equality holds in the right inequality if and only ifKis also origin-symmetric.
Theorems 1.1 and 1.2 belong to a part of new and rapidly evolving asymmetric \(L_{p}\) Brunn-Minkowski theory that has its origins in the work of Ludwig, Haberl and Schuster (see [4‐9]). For the studies of asymmetric \(L_{p}\) Brunn-Minkowski theory, also see [10‐22].
Haberl and Ludwig [5] defined the \(L_{p}\)-intersection body as follows: For \(K\in{\mathcal{S}}^{n}_{o}\), \(0< p<1\), the \(L_{p}\)-intersection body \(I_{p}K\) of K is the origin-symmetric star body whose radial function is given by
for all \(u\in{S}^{n-1}\). Haberl and Ludwig [5] pointed out that the classical intersection body which was introduced by Lutwak (see [3]) IK of K is obtained as a limit of the \(L_{p}\)-intersection body of K, more precisely, for all \(u\in{S}^{n-1}\),
Associated with the \(L_{p}\)-intersection bodies, Haberl [4] obtained a series of results, Berck [23] investigated their convexity. For further results on \(L_{p}\)-intersection bodies, also see [1, 2, 18, 24‐27]. In particular, Yuan and Cheung (see [26]) gave the negative solutions of \(L_{p}\)-Busemann-Petty problems as follows.
Theorem 1.A
Let\(K\in\mathcal{S}^{n}_{o}\)and\(0< p<1\), ifKis not origin-symmetric, then there exists\(L\in\mathcal{S}^{n}_{os}\)such that
$$I_{p}K\subset I_{p}L, $$
but
$$V(K)> V(L). $$
As the application of Theorem 1.1, we extend the scope of negative solutions of \(L_{p}\)-Busemann-Petty problems from origin-symmetric star bodies to star bodies.
Theorem 1.3
Let\(K\in\mathcal{S}^{n}_{o}\)and\(0< p<1\), ifKis not origin-symmetric, then there exists\(L\in\mathcal{S}^{n}_{o}\)such that
$$I_{p}K\subset I_{p}L, $$
but
$$V(K)> V(L). $$
Similarly, applying Theorem 1.2, we get the form of \(L_{p}\)-dual affine surface areas for the negative solutions of \(L_{p}\)-Busemann-Petty problems.
Theorem 1.4
For\({K}\in{\mathcal{S}}_{o}^{n}\), \(0< p<1\), ifKis not origin-symmetric, then there exists\({L}\in{\mathcal{S}}_{o}^{n}\)such that
In this paper, the proofs of Theorems 1.1-1.4 will be given in Section 4. In Section 3, we obtain some properties of general \(L_{p}\)-dual Blaschke bodies.
2 Preliminaries
2.1 \(L_{p}\)-Dual mixed volume
For \({K, L}\in S_{o}^{n}\), \(p > 0\) and \({\lambda, \mu} \geq0\) (not both zero), the \(L_{p}\)-radial combination, \(\lambda\cdot K\, \tilde{+}_{p}\, \mu\cdot L\in S_{o}^{n}\), of K and L is defined by (see [4, 28])
where \(\lambda\cdot K \) denotes the \(L_{p}\)-radial scalar multiplication, and we easily know \(\lambda\cdot K=\lambda^{\frac{1}{p}}K\).
Associated with (2.1), Haberl in [4] (also see [28]) introduced the notion of \(L_{p}\)-dual mixed volume as follows: For \({K, L}\in{\mathcal{S}}_{o}^{n}\), \(p > 0\), \(\varepsilon> 0\), the \(L_{p}\)-dual mixed volume \(\widetilde{V}_{p}(K, L)\) of K and L is defined by
The notion of \(L_{p}\)-dual affine surface area was given by Wang, Yuan and He (see [29]). For \(K\in{\mathcal{S}}_{o}^{n}\), \(0< p< n\), the \(L_{p}\)-dual affine surface area \(\widetilde{\Omega}_{p}(K)\) of K is defined by
Here \(E^{\ast}\) is the polar set of a nonempty set E which is defined by (see [1])
$$E^{\ast}=\bigl\{ x\in\mathbb{R}^{n}: x\cdot y\leq1 \mbox{ for all } y\in E\bigr\} . $$
For the sake of convenience of our work, we improve definition (2.6) from \(Q\in{\mathcal{K}}_{c}^{n}\) to \(Q\in{\mathcal{S}}_{os}^{n}\) as follows: For \(K\in{\mathcal{S}}_{o}^{n}\), \(0< p< n\), the \(L_{p}\)-dual affine surface area \(\widetilde{\Omega}_{p}(K)\) of K is defined by
Since \(f_{1}(\tau)-f_{2}(\tau) \neq0\) when \(\tau\neq0\), thus it follows that \(\rho(K, u)=\rho(-K, u)\) for all \(u \in S^{n-1}\), i.e., \(K\in{\mathcal{S}}_{os}^{n}\). □
From Theorem 3.2, it immediately yields the following corollary.
Corollary 3.1
For\(K \in{\mathcal{S}}_{o}^{n}\), \(n>p>0\)and\(\tau\in[-1, 1]\), ifKis not origin-symmetric, then\(\overline{\nabla}_{p}^{\tau}K=\overline{\nabla}_{p}^{-\tau} K\)if and only if\(\tau=0\).
Theorem 3.3
If\(K \in{\mathcal{S}}_{os}^{n}\), \(n>p>0\)and\(\tau\in[-1, 1]\), then
$$\overline{\nabla}_{p}^{\tau}K=K. $$
Proof
Since \(K \in{\mathcal{S}}_{os}^{n}\), i.e., \(K=-K\), by (1.3) and (1.5) we know that, for any \(u\in S^{n-1}\),
$$\rho\bigl(\overline{\nabla}_{p}^{\tau}K, u \bigr)^{n-p}=f_{1}(\tau)\rho(K, u)^{n-p}+f_{2}( \tau) \rho(-K, u)^{n-p}=\rho(K, u)^{n-p}. $$
That is,
$$\overline{\nabla}_{p}^{\tau}K=K. $$
□
4 Proofs of theorems
In this section, we complete the proofs of Theorems 1.1-1.4.
Lemma 4.1
If\(K, L \in\mathcal{S}_{o}^{n}\), \({\lambda, \mu \geq0}\) (not both zero), \(n > p>0\), then
$$ V(\lambda\circ K \oplus_{p} \mu\circ L)^{\frac{n-p}{n}} \leq\lambda V(K)^{\frac{n-p}{n}}+\mu V(L)^{\frac{n-p}{n}}, $$
(4.1)
with equality if and only ifKandLare dilates.
Proof
Associated with (1.1), (2.2), (2.3) and inequality (2.4), we know that, for any \(Q \in S_{o}^{n}\),
Let \(Q=\lambda\circ K\oplus_{p} \mu\circ L\), it yields (4.1). From the equality condition of (2.4), we see that equality holds in (4.1) if and only if K is a dilate of L. □
Clearly, equality holds in (4.2) if \(\tau=\pm1\). Besides, if \(\tau \neq\pm1\), then by the condition of equality in (4.1), we see that equality holds in (4.2) if and only if K and −K are dilates, this yields \(K=-K\), i.e., K is an origin-symmetric star body. This means that if \(\tau\neq\pm1\), then equality holds in the right inequality of (1.8) if and only if K is origin-symmetric.
Now, we prove the left inequality of (1.8). From (1.2), (1.4) and (1.7), we know that for any \(u\in S^{n-1}\),
$$\begin{aligned} \overline{\nabla}_{p} K =&\frac{1}{2}\circ K \oplus_{p} \frac{1}{2}\circ (-K) \\ =&\frac{1}{2}\frac{(1+\tau)+(1-\tau)}{2}\circ K \oplus_{p} \frac {1}{2}\frac{(1-\tau)+(1+\tau)}{2}\circ(-K) \\ =&\frac{1}{2}\circ\overline{\nabla}^{\tau}_{p} K \oplus_{p}\frac {1}{2}\circ\overline{\nabla}^{-\tau}_{p} K. \end{aligned}$$
(4.3)
From Theorem 3.1 and (4.3), use (4.1) to yield that for \(n>p>0\),
$$\begin{aligned} \begin{aligned} V(\overline{\nabla}_{p} K)^{\frac{n-p}{n}}&=V\biggl(\frac{1}{2} \circ \overline{\nabla}^{\tau}_{p} K\oplus_{p} \frac{1}{2}\circ\overline {\nabla}^{-\tau}_{p} K \biggr)^{\frac{n-p}{n}} \\ &\leq\frac{1}{2}V\bigl(\overline{\nabla}^{\tau}_{p} K \bigr)^{\frac {n-p}{n}}+\frac{1}{2}V\bigl(\overline{\nabla}^{-\tau}_{p} K\bigr)^{\frac {n-p}{n}} \\ &=\frac{1}{2}V\bigl(\overline{\nabla}^{\tau}_{p} K \bigr)^{\frac{n-p}{n}}+\frac {1}{2}V\bigl(-\overline{\nabla}^{\tau}_{p} K\bigr)^{\frac{n-p}{n}} \\ &=V\bigl(\overline{\nabla}^{\tau}_{p} K\bigr)^{\frac{n-p}{n}}. \end{aligned} \end{aligned}$$
Obviously, if \(\tau=0\), then equality holds in (4.4). If \(\tau\neq 0\), according to the equality condition of (4.1), we see that equality holds in (4.4) if and only if \(\widehat{\nabla}^{\tau}_{p} K\) and \(\overline{\nabla}^{-\tau}_{p} K\) are dilates, this implies \(\overline{\nabla}^{\tau}_{p} K=\overline{\nabla}^{-\tau}_{p} K\). Therefore, using Corollary 3.1, we obtain that if K is not an origin-symmetric body, then equality holds in (4.4) if and only if \(\tau=0\). This shows that if \(\tau\neq0\), then equality holds in the left inequality of (1.8) if and only if K is origin-symmetric. □
If \(\tau\neq\pm1\), equality of (4.5) holds if and only if K and −K are dilates. This yields \(K=-K\), thus K is an origin-symmetric star body. Since (4.5) and (4.6) are equivalent, hence equality holds in (4.6) if and only if K is an origin-symmetric star body when \(\tau \neq\pm1\). Therefore, if \(\tau\neq\pm1\), equality holds in the right inequality of (1.9) if and only if K is origin-symmetric.
Further, we complete the proof of the left inequality of (1.9). From Theorem 3.1, we know that
$$\overline{\nabla}^{-\tau}_{p} K=-\overline{ \nabla}^{\tau}_{p} K. $$
Similar to the proof of equality in inequality (4.6), we easily know that equality holds in (4.7) if and only if \(\overline{\nabla}^{\tau}_{p} K=\overline{\nabla}^{-\tau}_{p} K\) when \(\tau\neq0\). Hence, if \(\tau\neq0\), using Theorem 3.2 we get that equality holds in the left inequality of (1.9) if and only if K is origin-symmetric. □
In order to prove Theorems 1.3 and 1.4, the following lemma is required.
Lemma 4.2
If\(K\in S_{o}^{n}\), \(0< p<1\)and\(\tau\in[-1,1]\), then
$$I_{p}\bigl(\overline{\nabla}^{\tau}_{p} K \bigr)=I_{p}K. $$
Proof
From definition (1.10), we may obtain the following polar coordinate form:
Choose \(\varepsilon>0\) such that \(V ((1+\varepsilon)\overline{\nabla}^{\tau}_{p} K )< V(K)\). Therefore, let \(L=(1+\varepsilon)\overline{\nabla}^{\tau}_{p} K\) (for \(\tau=0\), \(L\in S^{n}_{os}\); for \(\tau\neq0\), \(L\in S^{n}_{o}\)), then
$$V(L)< V(K). $$
But from Lemma 4.2, and notice that \(I_{p} ((1+\varepsilon)K )=(1+\varepsilon)^{\frac{n-p}{p}}I_{p}K\), we can get
Choose \(\varepsilon>0\) such that \(\widetilde{\Omega}_{p} ((1+\varepsilon)\overline{\nabla }^{\tau}_{p} K )<\widetilde{\Omega}_{p}(K)\). Therefore, let \(L=(1+\varepsilon)\overline{\nabla}^{\tau}_{p} K\), then \(L\in S^{n}_{o}\) and
But, similar to the proof of Theorem 1.3, we may obtain \(I_{p}L\supset I_{p} K\). □
Acknowledgements
The authors would like to sincerely thank the referees for very 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), Innovation Foundation and Excellent Foundation of Graduate Student of China Three Gorges University (Grant Nos. 2013CX084, 2014PY065).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.