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.

Theorem 1.1

If \(K \in{\mathcal{S}}_{o}^{n}\), \(n>p> 0\), \(\tau\in[-1, 1]\), then

$$ V(\overline{\nabla}_{p} K)\leq V\bigl(\overline{\nabla}_{p}^{\tau}K\bigr)\leq V(K). $$
(1.8)

If \(\tau\neq0\), equality holds in the left inequality if and only if K is origin-symmetric, if \(\tau\neq\pm 1\), then equality holds in the right inequality if and only if K is 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

$$ \widetilde{\Omega}_{p}(\overline{\nabla}_{p} K) \leq \widetilde {\Omega}_{p}\bigl(\overline{\nabla}_{p}^{\tau}K\bigr) \leq\widetilde{\Omega }_{p}(K). $$
(1.9)

If \(\tau\neq0\), equality holds in the left inequality if and only if K is origin-symmetric, if \(\tau\neq\pm 1\), then equality holds in the right inequality if and only if K is 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

$$ \rho^{p}_{I_{p}K}(u)=\int_{K}|u\cdot x|^{-p}\, dx $$
(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.

Theorem 1.A

Let \(K\in\mathcal{S}^{n}_{o}\) and \(0< p<1\), if K is 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 \(0< p<1\), if K is 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 \({K}\in{\mathcal{S}}_{o}^{n}\), \(0< p<1\), if K is not origin-symmetric, then there exists \({L}\in{\mathcal{S}}_{o}^{n}\) such that

$$I_{p}K\subset I_{p}L, $$

but

$$\widetilde{\Omega}_{p}(K) > \widetilde{\Omega}_{p}(L). $$

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])

$$ \rho(\lambda\cdot K \, \tilde{+}_{p}\, \mu\cdot L, \cdot)^{p} = \lambda \rho(K, \cdot)^{p} +\mu \rho(L, \cdot)^{p}, $$
(2.1)

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

$$\frac{n}{p}\widetilde{V}_{p}(K, L)=\lim_{\varepsilon\rightarrow 0^{+}} \frac{V(K\, \tilde{+}_{p}\, \varepsilon\cdot L)-V(K)}{\varepsilon}. $$

And he got the following integral form of \(L_{p}\)-dual mixed volume:

$$ \widetilde{V}_{p}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho _{K}^{n-p}(u)\rho_{L}^{p}(u)\, du, $$
(2.2)

where the integration is with respect to spherical Lebesgue measure on \(S^{n-1}\).

From (2.2), we get that

$$ \widetilde{V}_{p}(K, K)=V(K)=\frac{1}{n}\int _{S^{n-1}}\rho _{K}^{n}(u)\, du. $$
(2.3)

The Minkowski inequality of \(L_{p}\)-dual mixed volume is as follows (see [4, 28]): If \({K,L}\in S_{o}^{n}\), then for \(0< p < n\),

$$ \widetilde{V}_{p}(K, L)\leq V(K)^{\frac{n-p}{n}}V(L)^{\frac {p}{n}}; $$
(2.4)

for \(p> n\),

$$ \widetilde{V}_{p}(K, L)\geq V(K)^{\frac{n-p}{n}}V(L)^{\frac {p}{n}}. $$
(2.5)

In every case, equality holds if and only if K is a dilate of L. For \(p=n\), (2.4) (or (2.5)) is identical.

From (2.4) and (2.5), we easily get the following result.

Proposition 2.1

If \(K, L\in{\mathcal{S}}_{o}^{n}\), \(p> 0\), and for any \(Q\in{\mathcal{S}}_{o}^{n}\),

$$\widetilde{V}_{p}(K, Q)=\widetilde{V}_{p}(L, Q) $$

or

$$\widetilde{V}_{p}(Q, K)=\widetilde{V}_{p}(Q, L), $$

then

$$K=L. $$

2.2 \(L_{p}\)-Dual affine surface area

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

$$ n^{-\frac{p}{n}}\widetilde{\Omega}_{p}(K)^{\frac{n+p}{n}}= \sup\bigl\{ n\widetilde{V}_{p}\bigl(K,Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q \in{\mathcal{K}}_{c}^{n}\bigr\} . $$
(2.6)

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

$$ n^{-\frac{p}{n}}\widetilde{\Omega}_{p}(K)^{\frac{n+p}{n}}= \sup\bigl\{ n\widetilde{V}_{p}\bigl(K,Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q \in{\mathcal{S}}_{os}^{n}\bigr\} . $$
(2.7)

3 Some properties of general \(L_{p}\)-dual Blaschke bodies

In this section, we give some properties of general \(L_{p}\)-dual Blaschke bodies.

Theorem 3.1

If \(K \in{\mathcal{S}}_{o}^{n}\), \(n>p>0\) and \(\tau\in[-1, 1]\), then

$$\overline{\nabla}_{p}^{-\tau}K=\overline{\nabla}_{p}^{\tau }(-K)=- \overline{\nabla}_{p}^{\tau}K. $$

Proof

From (1.6) and (1.7), we obtain that for \(n>p>0\) and \(\tau \in[-1, 1]\),

$$\overline{\nabla}_{p}^{-\tau}K=f_{1}(-\tau)\circ K \oplus_{p}f_{2}(-\tau )\circ(-K) =f_{2}(\tau)\circ K\oplus_{p}f_{1}(\tau)\circ(-K)=\overline{\nabla }^{\tau}_{p} (-K). $$

Further, we have that for any \(u\in S^{n-1}\),

$$\begin{aligned} \rho\bigl(-\overline{\nabla}^{\tau}_{p} K, u \bigr)^{n-p} =&\rho\bigl(\overline {\nabla}^{\tau}_{p} K, -u\bigr)^{n-p} \\ =&f_{1}(\tau)\rho(K, -u)^{n-p}+f_{2}(\tau) \rho(-K, -u)^{n-p} \\ =&f_{1}(\tau)\rho(-K, u)^{n-p}+f_{2}(\tau)\rho \bigl(-(-K), u\bigr)^{n-p} \\ =&\rho\bigl(\overline{\nabla}^{\tau}_{p} (-K), u \bigr)^{n-p}. \end{aligned}$$

Hence, we get

$$\overline{\nabla}_{p}^{\tau}(-K)=-\overline{ \nabla}_{p}^{\tau}K. $$

 □

Theorem 3.2

For \(K \in{\mathcal{S}}_{o}^{n}\), \(n>p>0\) and \(\tau\in[-1, 1]\), if \(\tau\neq0\), then \(\overline{\nabla}^{\tau}_{p} K=\overline{\nabla}^{-\tau}_{p} K\) if and only if \(K\in{\mathcal{S}}_{os}^{n}\).

Proof

From (1.3) and (1.6), we get that for all \(u \in S^{n-1}\),

$$\begin{aligned}& \rho\bigl(\overline{\nabla}_{p}^{\tau}K, u \bigr)^{n-p}=f_{1}(\tau) \rho(K, u)^{n-p} + f_{2}(\tau) \rho(-K, u)^{n-p}, \end{aligned}$$
(3.1)
$$\begin{aligned}& \rho\bigl(\overline{\nabla}_{p}^{-\tau} K, u \bigr)^{n-p}=f_{2}(\tau) \rho(K, u)^{n-p}+f_{1}( \tau)\rho(-K, u)^{n-p}. \end{aligned}$$
(3.2)

Hence, if \(K\in{\mathcal{S}}_{os}^{n}\), i.e., \(K=-K\), then by (3.1), (3.2) and (1.5) we get, for all \(u\in S^{n-1}\),

$$\rho\bigl(\overline{\nabla}_{p}^{\tau}K, u \bigr)^{n-p}=\rho\bigl(\overline{\nabla }_{p}^{-\tau} K, u\bigr)^{n-p}. $$

Thus

$$\overline{\nabla}_{p}^{\tau}K=\overline{\nabla}_{p}^{-\tau} K. $$

Conversely, if \(\overline{\nabla}_{p}^{\tau}K= \overline{\nabla }_{p}^{-\tau} K\), then together with (3.1) and (3.2) it yields

$$\bigl[f_{1}(\tau)-f_{2}(\tau)\bigr] \rho(K, u)^{n-p}=\bigl[f_{1}(\tau)-f_{2}(\tau)\bigr] \rho (-K, u)^{n-p}. $$

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 \(\tau\in[-1, 1]\), if K is 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 if K and L are dilates.

Proof

Associated with (1.1), (2.2), (2.3) and inequality (2.4), we know that, for any \(Q \in S_{o}^{n}\),

$$\begin{aligned} \widetilde{V}_{p}(\lambda\circ K\oplus_{p}\mu\circ L, Q) =& \lambda \widetilde{V}_{p}(K,Q)+\mu\widetilde{V}_{p}(L,Q) \\ \leq& \bigl[\lambda V(K)^{\frac{n-p}{n}}+\mu V(L)^{\frac{n-p}{n}}\bigr]V(Q)^{\frac{p}{n}}. \end{aligned}$$

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. □

Proof of Theorem 1.1

By (4.1), (1.5) and (1.7), we get, for any \(\tau\in[-1, 1]\),

$$\begin{aligned} V\bigl(\overline{\nabla}_{p}^{\tau}K\bigr)^{\frac{n-p}{n}} =&V \bigl(f_{1}(\tau)\circ K\oplus_{p}f_{2}(\tau) \circ(-K)\bigr)^{\frac{n-p}{n}} \\ \leq& f_{1}(\tau)V(K)^{\frac{n-p}{n}}+f_{2}( \tau)V(-K)^{\frac{n-p}{n}} \\ =&V(K)^{\frac{n-p}{n}}. \end{aligned}$$

Therefore, we obtain, for \(n>p>0\),

$$ V\bigl(\overline{\nabla}_{p}^{\tau}K\bigr)\leq V(K). $$
(4.2)

This gives the right inequality of (1.8).

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 \(n>p>0\),

$$\begin{aligned} \begin{aligned} V(\overline{\nabla}_{p} K)^{\frac{n-p}{n}}&=V\biggl(\frac{1}{2} \circ \overline{\nabla}^{\tau}_{p} K\oplus_{p} \frac{1}{2}\circ\overline {\nabla}^{-\tau}_{p} K \biggr)^{\frac{n-p}{n}} \\ &\leq\frac{1}{2}V\bigl(\overline{\nabla}^{\tau}_{p} K \bigr)^{\frac {n-p}{n}}+\frac{1}{2}V\bigl(\overline{\nabla}^{-\tau}_{p} K\bigr)^{\frac {n-p}{n}} \\ &=\frac{1}{2}V\bigl(\overline{\nabla}^{\tau}_{p} K \bigr)^{\frac{n-p}{n}}+\frac {1}{2}V\bigl(-\overline{\nabla}^{\tau}_{p} K\bigr)^{\frac{n-p}{n}} \\ &=V\bigl(\overline{\nabla}^{\tau}_{p} K\bigr)^{\frac{n-p}{n}}. \end{aligned} \end{aligned}$$

This gives that for \(n>p>0\),

$$ V(\overline{\nabla}_{p} K)\leq V\bigl(\overline{\nabla}^{\tau}_{p} K\bigr). $$
(4.4)

This is just the left inequality of (1.8).

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. □

Proof of Theorem 1.2

From definition (2.7) and (1.7), we have that

$$\begin{aligned}& n^{-\frac{p}{n}}\widetilde{\Omega}_{p}\bigl(\overline{ \nabla}^{\tau}_{p} K\bigr)^{\frac{n+p}{n}} \\& \quad =\sup \bigl\{ n\widetilde{V}_{p}\bigl(\widehat{ \nabla}^{\tau}_{p} K, Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q\in{\mathcal{S}}_{os}^{n} \bigr\} \\& \quad =\sup \bigl\{ n \widetilde{V}_{p} \bigl(f_{1}(\tau) \circ K\oplus _{p}f_{2}(\tau)\circ(-K), Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q\in {\mathcal{S}}_{os}^{n} \bigr\} \\& \quad =\sup \biggl\{ \int_{S^{n-1}} \bigl[\rho \bigl(f_{1}(\tau)\circ K\oplus_{p} f_{2}(\tau) \circ(-K), u\bigr)^{n-p}\rho\bigl(Q^{\ast},u\bigr)^{p} \bigr]\, duV(Q)^{\frac {p}{n}}: Q\in{\mathcal{S}}_{os}^{n} \biggr\} \\& \quad = \sup \biggl\{ \int_{S^{n-1}}\bigl[f_{1}(\tau) \rho(K, u)^{n-p}+ f_{2}(\tau) \rho(-K, u)^{n-p}\bigr] \rho\bigl(Q^{\ast},u\bigr)^{p}\, duV(Q)^{\frac{p}{n}}: Q\in{ \mathcal{S}}_{os}^{n} \biggr\} \\& \quad =\sup \bigl\{ nf_{1}(\tau)\widetilde{V}_{p}\bigl(K, Q^{\ast}\bigr)V(Q)^{\frac {p}{n}}+nf_{2}(\tau) \widetilde{V}_{p}\bigl(-K, Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q \in{\mathcal{S}}_{os}^{n} \bigr\} \\& \quad \leq f_{1}(\tau) \sup \bigl\{ n\widetilde{V}_{p} \bigl(K, Q^{\ast}\bigr)V(Q)^{\frac {p}{n}}: Q\in{\mathcal{S}}_{os}^{n} \bigr\} \\& \qquad {}+ f_{2}(\tau) \sup \bigl\{ n\widetilde{V}_{p} \bigl(-K, Q^{\ast}\bigr)V(Q)^{\frac {p}{n}}: Q\in{\mathcal{S}}_{os}^{n} \bigr\} . \end{aligned}$$
(4.5)

Since \(Q\in{\mathcal{S}}_{os}^{n}\), thus use \(\rho(Q, u)=\rho(-Q, u)=\rho(Q, -u)\) for all \(u\in S^{n-1}\) to get

$$\widetilde{V}_{p}\bigl(-K, Q^{\ast}\bigr)= \widetilde{V}_{p}\bigl(K, Q^{\ast}\bigr), $$

by (2.7) we know \(\widetilde{\Omega}_{p}(K)=\widetilde{\Omega}_{p}(-K)\). This combining with (4.5) and (1.5), we know

$$ \widetilde{\Omega}_{p}\bigl(\overline{\nabla}^{\tau}_{p} K\bigr) \leq \widetilde{\Omega}_{p}(K), $$
(4.6)

i.e., the right inequality of (1.9) is obtained.

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

$$\overline{\nabla}^{-\tau}_{p} K=-\overline{ \nabla}^{\tau}_{p} K. $$

Thus, (4.3) can be written as

$$\overline{\nabla}_{p} K=\frac{1}{2}\circ\overline{ \nabla}^{\tau}_{p} K\oplus_{p}\frac{1}{2}\circ \bigl(-\overline{\nabla}^{\tau}_{p} K\bigr). $$

Similar to the proof of inequality (4.6), we have

$$ \widetilde{\Omega}_{p}(\overline{\nabla}_{p} K) \leq \widetilde {\Omega}_{p}\bigl(\overline{\nabla}^{\tau}_{p} K\bigr). $$
(4.7)

This yields the left inequality of (1.9).

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:

$$\rho(I_{p}K, u)^{p}=\frac{1}{n-p}\int _{S^{n-1}}|u\cdot v|^{-p}\rho(K, v)^{n-p}\, dv. $$

Thus by (1.3) we have that

$$\begin{aligned} \rho \bigl(I_{p}\bigl(\overline{\nabla}^{\tau}_{p} K \bigr), u \bigr)^{p} =&\frac {1}{n-p}\int_{S^{n-1}}|u \cdot v|^{-p}\rho \bigl(\overline{\nabla }^{\tau}_{p} K, v \bigr)^{n-p} \, dv \\ =&\frac{1}{n-p}\int_{S^{n-1}}|u\cdot v|^{-p} \bigl[f_{1}(\tau)\rho (K,v)^{n-p}+f_{2}(\tau) \rho(-K,v)^{n-p} \bigr]\, dv \\ =&f_{1}(\tau)\rho(I_{p}K, u)^{p}+f_{2}( \tau)\rho \bigl(I_{p}(-K), u \bigr)^{p}. \end{aligned}$$
(4.8)

According to (1.10), we easily know \(I_{p}(-K)=I_{p}K\), so combining with (4.8) and (1.5), then for any \(u\in S^{n-1}\),

$$\rho \bigl(I_{p}\bigl(\overline{\nabla}^{\tau}_{p}K \bigr), u \bigr)^{p}=\rho(I_{p} K, u)^{p}, $$

i.e.,

$$I_{p}\bigl(\overline{\nabla}^{\tau}_{p}K \bigr)=I_{p}K. $$

 □

Proof of Theorem 1.3

Since K is not an origin-symmetric star body, thus from Theorem 1.1, we know that if \(\tau\neq\pm1\), then

$$V\bigl(\overline{\nabla}^{\tau}_{p} K\bigr)< V(K). $$

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

$$I_{p}L=I_{p} \bigl((1+\varepsilon)\overline{ \nabla}_{p}^{\tau}L \bigr)=(1+\varepsilon)^{\frac{n-p}{p}}I_{p} \bigl(\overline{\nabla}_{p}^{\tau}K\bigr) =(1+ \varepsilon)^{\frac{n-p}{p}}I_{p} K\supset I_{p} K. $$

 □

Proof of Theorem 1.4

Since K is not an origin-symmetric star body, thus by Theorem 1.2, we know that for \(\tau\neq\pm1\),

$$\widetilde{\Omega}_{p}\bigl(\overline{\nabla}^{\tau}_{p} K\bigr)< \widetilde {\Omega}_{p}(K). $$

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

$$\widetilde{\Omega}_{p}(L)< \widetilde{\Omega}_{p}(K). $$

But, similar to the proof of Theorem 1.3, we may obtain \(I_{p}L\supset I_{p} K\). □