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])
$$\rho(K,u)=\max\{\lambda\geq0: \lambda\cdot u\in K\},\quad u\in S^{n-1}. $$
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
\(u\in S^{n-1}\).
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
$$ \overline{\nabla}_{p}K=\frac{1}{2}\circ K\oplus_{p} \frac {1}{2}\circ(-K). $$
(1.2)
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
$$ \rho\bigl(\overline{\nabla}_{p}^{\tau}K, \cdot \bigr)^{n-p}=f_{1}(\tau)\rho(K, \cdot)^{n-p}+f_{2}( \tau)\rho(-K, \cdot)^{n-p}, $$
(1.3)
where
$$ f_{1}(\tau)=\frac{1+\tau}{2},\qquad f_{2}(\tau)= \frac{1-\tau}{2}. $$
(1.4)
From (
1.4), we have that
$$\begin{aligned}& f_{1}(\tau)+f_{2}(\tau)=1, \end{aligned}$$
(1.5)
$$\begin{aligned}& f_{1}(-\tau)=f_{2}(\tau),\qquad f_{2}(- \tau)=f_{1}(\tau). \end{aligned}$$
(1.6)
From (
1.3), it easily follows that
$$ \overline{\nabla}_{p}^{\tau}K=f_{1}(\tau)\circ K \oplus_{p} f_{2}(\tau )\circ(-K). $$
(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.
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.
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
$$ \rho^{p}_{I_{p}K}(u)=\int_{K}|u\cdot x|^{-p}\, dx $$
(1.10)
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
\(u\in{S}^{n-1}\),
$$ \rho(IK, u) = \lim_{p\longrightarrow1^{-}}(1-p)\rho(I_{p}K, u)^{p}. $$
(1.11)
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.
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.
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.
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.
3 Some properties of general \(L_{p}\)-dual Blaschke bodies
In this section, we give some properties of general \(L_{p}\)-dual Blaschke bodies.
From Theorem
3.2, it immediately yields the following corollary.
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.