Open Access

\(L_{p}\)-Dual geominimal surface area and general \(L_{p}\)-centroid bodies

Journal of Inequalities and Applications20152015:358

https://doi.org/10.1186/s13660-015-0888-9

Received: 24 August 2015

Accepted: 6 November 2015

Published: 14 November 2015

Abstract

In this article, we consider the Shephard type problems and obtain the affirmative and negative parts of the version of \(L_{p}\)-dual geominimal surface area for general \(L_{p}\)-centroid bodies. Combining with the \(L_{p}\)-dual geominimal surface area we also give a negative form of the Shephard type problems for \(L_{p}\)-centroid bodies.

Keywords

\(L_{p}\)-dual geominimal surface area general \(L_{p}\)-centroid bodies \(L_{p}\)-centroid bodies Shephard type problems

MSC

52A20 52A40

1 Introduction and main results

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 \(\mathcal{K}^{n}_{c}\), respectively. \(S^{n}_{o}\) and \(S^{n}_{c}\), respectively, denote the set of star bodies (about the origin) and the set of origin-symmetric star bodies 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 a body K. For the standard unit ball B in \(\mathbb{R}^{n}\), we use \(\omega_{n} = V(B)\) to denote its volume.

The notion of geominimal surface area was discovered by Petty (see [1]). For \(K\in\mathcal{K}^{n}\), the geominimal surface area, \(G(K)\), of K is defined by
$$\omega_{n}^{\frac{1}{n}}G(K)= \inf\bigl\{ nV_{1}(K,Q)V \bigl(Q^{\ast}\bigr)^{\frac{1}{n}}: Q\in\mathcal{K}^{n}\bigr\} . $$
Here \(Q^{\ast}\) denotes the polar of body Q and \(V_{1}(M, N)\) denotes the mixed volume of \(M, N\in\mathcal{K}^{n}\) (see [2]).

The geominimal surface area serves as a bridge connecting a number of areas of geometry: affine differential geometry, relative geometry, and Minkowskian geometry. Hence it receives a lot of attention (see, e.g., [3, 4]). Lutwak in [5] showed that there were natural extensions of geominimal surface areas in the Brunn-Minkowski-Firey theory. It motivates extensions of some known inequalities for geominimal surface areas to \(L_{p}\)-geominimal surface areas. The inequalities for \(L_{p}\)-geominimal surface areas are stronger than their classical counterparts (see [610]).

Based on \(L_{p}\)-mixed volume, Lutwak [5] introduced the notion of \(L_{p}\)-geominimal surface area. For \(K\in\mathcal{K}^{n}_{o}\), \(p\geq1\), the \(L_{p}\)-geominimal surface area, \(G_{p}(K)\), of K is defined by
$$\omega_{n}^{\frac{p}{n}}G_{p}(K)= \inf\bigl\{ nV_{p}(K,Q)V\bigl(Q^{\ast}\bigr)^{\frac{p}{n}}: Q\in \mathcal{K}^{n}_{o}\bigr\} . $$
Here \(V_{p}(M, N)\) denotes the \(L_{p}\)-mixed volume of \(M, N\in\mathcal{K}^{n}_{o}\) (see [5, 11]). Obviously, if \(p=1\), \(G_{p}(K)\) is just the geominimal surface area \(G(K)\).
Recently, Wang and Qi [12] introduced a concept of \(L_{p}\)-dual geominimal surface area, which is a dual concept for \(L_{p}\)-geominimal surface area and belongs to the dual \(L_{p}\)-Brunn-Minkowski theory for star bodies also developed by Lutwak (see [13, 14]). For \(K\in S_{o}^{n}\), and \(p\geq1\), the \(L_{p}\)-dual geominimal surface area, \(\widetilde{G}_{-p}(K)\), of K is defined by
$$ \omega_{n}^{-\frac{p}{n}}\widetilde{G}_{-p}(K)= \inf\bigl\{ n \widetilde{V}_{-p}(K,Q)V\bigl(Q^{\ast}\bigr)^{-\frac{p}{n}}: Q\in \mathcal{K}_{c}^{n}\bigr\} . $$
(1.1)
Here, \(\widetilde{V}_{-p}(M,N)\) denotes the \(L_{p}\)-dual mixed volume of \(M, N\in S_{o}^{n}\) (see [5]).
Centroid bodies are a classical notion from geometry which have attracted increased attention in recent years (see [13, 1522]). In particular, Lutwak and Zhang [18] introduced the notion of \(L_{p}\)-centroid bodies. For each compact star-shaped (about the origin) K in \(\mathbb{R}^{n}\) and real number \(p\geq1\), the \(L_{p}\)-centroid body, \(\Gamma_{p} K\), of K is an origin-symmetric convex body whose support function is defined by
$$\begin{aligned} h^{p}_{\Gamma_{p} K}(u)&=\frac{1}{c_{n, p}V(K)}\int _{K}|u\cdot x|^{p} \,dx \\ &=\frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}|u\cdot v|^{p} \rho_{K}^{n+p}(v)\,dS(v) \end{aligned}$$
(1.2)
for all \(u\in S^{n-1}\), where
$$ c_{n, p}=\omega_{n+p}/\omega_{2}\omega_{n} \omega_{p-1}, \quad\mbox{and}\quad \omega _{n}= \pi^{\frac{n}{2}}/\Gamma\biggl(1+\frac{n}{2}\biggr). $$
(1.3)
More recently, Feng et al. [23] defined a new notion of general \(L_{p}\)-centriod bodies, which generalized the concept of \(L_{p}\)-centroid bodies. For \(K\in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), the general \(L_{p}\)-centroid body, \(\Gamma_{p}^{\tau}K\), of K is a convex body whose support function is defined by
$$\begin{aligned} h^{p}_{\Gamma_{p}^{\tau}K}(u)&=\frac{1}{c_{n, p}(\tau)V(K)}\int _{K}\varphi _{\tau}(u\cdot x)^{p} \,dx \\ & =\frac{1}{c_{n, p}(\tau)(n+p)V(K)}\int_{S^{n-1}}\varphi_{\tau}(u\cdot v)^{p}\rho_{K}^{n+p}(v)\,dv, \end{aligned}$$
(1.4)
where
$$c_{n, p}(\tau)=\frac{1}{2}c_{n, p}\bigl[(1+ \tau)^{p}+(1-\tau)^{p}\bigr], $$
and \(\varphi_{\tau}: \mathbb{R}\rightarrow[0, \infty)\) is a function defined by \(\varphi_{\tau}(t)=|t|+\tau t\). We note that general \(L_{p}\)-centroid bodies are an essential part of the rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [20, 2432]).
The normalization is chosen such that \(\Gamma_{p}^{\tau}B=B\) for every \(\tau\in[-1, 1]\), and \(\Gamma_{p}^{0} K=\Gamma_{p} K\). Let \(\varphi_{+}(u\cdot x)=\max\{u\cdot x, 0\}\) (\(\tau=1\)) in (1.4), then a special case of the definition of \(\Gamma_{p}^{\tau}K\) is \(\Gamma_{p}^{+} K\), i.e.,
$$\begin{aligned} h^{p}_{\Gamma_{p}^{+} K}(u)&=\frac{1}{c_{n, p}V(K)}\int _{K}\varphi_{+}(u\cdot x)^{p} \,dx \\ &=\frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p}\rho_{K}^{n+p}(v)\,dv. \end{aligned}$$
(1.5)
Besides, we also define
$$ \Gamma_{p}^{-}K=\Gamma_{p}^{+}(-K). $$
(1.6)
From the definition of \(\Gamma^{\pm}_{p} K\) and (1.4), we see that if \(K\in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
$$ \Gamma_{p}^{\tau}K=f_{1}(\tau)\cdot \Gamma_{p}^{+} K+_{p}f_{2}(\tau)\cdot \Gamma_{p}^{-} K, $$
(1.7)
where ‘\(+_{p}\)’ denotes the Firey \(L_{p}\)-combination of convex bodies, and
$$ f_{1}(\tau)=\frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}, \qquad f_{2}(\tau )= \frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$
(1.8)
If \(\tau=\pm1\) in (1.7) and using (1.8), then
$$\Gamma_{p}^{+1} K=\Gamma_{p}^{+} K, \qquad \Gamma_{p}^{-1} K=\Gamma_{p}^{-} K. $$
In [16] Grinberg and Zhang discussed an investigation of Shephard type problems for \(L_{p}\)-centriod bodies. Namely, let K and L be two origin-symmetric star bodies such that
$$\Gamma_{p} K\subset\Gamma_{p} L. $$
They proved that if the space \((\mathbb{R}^{n}, \|\cdot\|_{L})\) embeds in \(L_{p}\), then we necessarily have
$$V(K)\leq V(L). $$
On the other hand, if \((\mathbb{R}^{n}, \|\cdot\|_{K})\) does not embed in \(L_{p}\), then there is a body L so that \(\Gamma_{p} K\subset\Gamma_{p} L\), but \(V(K)\leq V(L)\).

In this article, we first investigate the Shephard type problems for general \(L_{p}\)-centroid bodies and give the affirmative and negative parts of the version of \(L_{p}\)-dual geominimal surface area.

Theorem 1.1

For \(K \in\mathcal{K}_{o}^{n}\), \(L\in\mathcal{K}_{c}^{n}\), and \(p\geq1\), if \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), then
$$ \widetilde{G}_{-p}(K)\leq\widetilde{G}_{-p}(L), $$
(1.9)
with equality if and only if \(K=L\).

Theorem 1.2

For \(L \in S_{o}^{n}\), \(p\geq1\) and \(\tau\in (-1, 1)\), if L is not origin-symmetric, then there exists \(K\in S_{o}^{n}\), such that
$$\Gamma_{p}^{+} K\subset\Gamma_{p}^{\tau}L,\qquad \Gamma_{p}^{-} K\subset\Gamma _{p}^{-\tau} L. $$
But
$$\widetilde{G}_{-p}(K)>\widetilde{G}_{-p}(L). $$

Further, taking together the \(L_{p}\)-dual geominimal surface area with \(L_{p}\)-centroid bodies we establish the following Shephard type problem.

Theorem 1.3

For \(L \in S_{o}^{n}\) and \(1\leq p< n\), if L is not origin-symmetric star body, then there exists \(K\in S_{o}^{n}\), such that
$$\Gamma_{p}K\subset\Gamma_{p} L. $$
But
$$\widetilde{G}_{-p}(K)>\widetilde{G}_{-p}(L). $$

The proofs of Theorems 1.1-1.3 will be given in Section 3.

2 Preliminaries

2.1 Support functions, radial functions, and polars of convex bodies

The support function, \(h_{K} = h(K,\cdot):\mathbb{R}^{n}\rightarrow(-\infty ,\infty)\), of \(K\in\mathcal{K}^{n}\) is defined by (see [33, 34])
$$ h(K,x)=\max\{x \cdot y: y\in K\},\quad x\in\mathbb{R}^{n}, $$
(2.1)
where \(x\cdot y\) denotes the standard inner product of x and y.
If K is a compact star-shaped (about the origin) set in \(\mathbb {R}^{n}\), then its radial function, \(\rho_{K}=\rho(K,\cdot):\mathbb{R}^{n}\setminus\{0\}\rightarrow[0,\infty )\), is defined by (see [33, 34])
$$ \rho(K,u)=\max\{\lambda\geq0: \lambda\cdot u\in K\},\quad u\in S^{n-1}. $$
(2.2)
If \(\rho_{K}\) is continuous and positive, then K will be called a star body. Two star bodies K, L are said to be dilates (of one another) if \(\rho_{K}(u)\diagup\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
If \(K\in\mathcal{K}_{o}^{n}\), the polar body, \(K^{\ast}\), of K is defined by (see [33, 34])
$$ K^{\ast}=\bigl\{ x\in\mathbb{R}^{n}: x\cdot y\leq1,y\in K\bigr\} . $$
(2.3)
For \(K, L\in\mathcal{K}_{o}^{n}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the Firey \(L_{p}\)-combination, \(\lambda\cdot K +_{p}\mu\cdot L\), of K and L is defined by (see [35])
$$ h(\lambda\cdot K +_{p}\mu\cdot L, \cdot)^{p}=\lambda h(K, \cdot)^{p}+\mu h(L, \cdot)^{p}, $$
(2.4)
where ‘  ’ in \(\lambda\cdot K\) denotes the Firey scalar multiplication. Obviously, the \(L_{p}\)-Firey and the usual scalar multiplications are related by \(\lambda\cdot K=\lambda^{\frac{1}{p}}K\).
For \({K, L}\in S_{o}^{n}\), \(p\geq1\), and \({\lambda, \mu} \geq0\) (not both zero), the \(L_{p}\)-harmonic radial combination, \(\lambda\star K +_{-p} \mu\star L\in S_{o}^{n}\), of K and L is defined by (see [5])
$$ \rho(\lambda\star K +_{-p}\mu\star L, \cdot)^{-p} = \lambda \rho(K, \cdot )^{-p} +\mu \rho(L, \cdot)^{-p}, $$
(2.5)
where \(\lambda\star K\) denotes the \(L_{p}\)-harmonic radial scalar multiplication. Here, we have \(\lambda\star K=\lambda^{-\frac{1}{p}}K\).

2.2 \(L_{p}\)-Dual mixed volume

Using \(L_{p}\)-harmonic radial combination, Lutwak [5] introduced the notion of \(L_{p}\)-dual mixed volume. For \({K, L}\in S_{o}^{n}\), \(p \geq1\), and \(\varepsilon> 0\), the \(L_{p}\)-dual mixed volume, \(\widetilde{V}_{-p}(K, L)\), of K and L is defined by
$$\frac{n}{-p}\widetilde{V}_{-p}(K, L)=\lim_{\varepsilon\rightarrow 0^{+}} \frac{V(K+_{-p}\varepsilon\star L)-V(K)}{\varepsilon}. $$
The definition above and de l’Hospital’s rule yield the following integral representation of \(L_{p}\)-dual mixed volume (see [5]):
$$ \widetilde{V}_{-p}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho_{K}^{n+p}(u)\rho _{L}^{-p}(u)\,du, $$
(2.6)
where the integration is with respect to spherical Lebesgue measure on \(S^{n-1}\).
From (2.6), it follows immediately that, for each \(K\in S_{o}^{n}\) and \(p\geq1\),
$$ \widetilde{V}_{-p}(K, K)=V(K)=\frac{1}{n}\int _{S^{n-1}}\rho _{K}^{n}(u)\,du. $$
(2.7)

Minkowski’s inequality for a \(L_{p}\)-dual mixed volume can be stated as follows (see [5]).

Theorem 2.A

If \({K, L}\in S_{o}^{n}\), \(p \geq1\), then
$$ \widetilde{V}_{-p}(K, L)\geq V(K)^{\frac{n+p}{n}}V(L)^{-\frac{p}{n}}, $$
(2.8)
with equality if and only if K and L are dilates.

2.3 General \(L_{p}\)-harmonic Blaschke bodies

For \(K\in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), the general \(L_{p}\)-harmonic Blaschke body, \(\widehat{{\nabla}}_{p}^{\tau}K\), of K is defined by (see [36])
$$ \frac{\rho(\widehat{\nabla}_{p}^{\tau}K, \cdot)^{n+p}}{V(\widehat{\nabla }_{p}^{\tau}K)}=f_{1}(\tau)\frac{\rho(K, \cdot)^{n+p}}{V(K)}+f_{2}( \tau)\frac {\rho(-K, \cdot)^{n+p}}{V(-K)}. $$
(2.9)

Operators of this type and related maps compatible with linear transformations appear essentially in the theory of valuations in connection with isoperimetric and analytic inequalities (see [3743]).

Theorem 2.B

[36]

If \(K \in S_{o}^{n}\), \(p\geq1\), and \(\tau\in(-1, 1)\), then
$$ \widetilde{G}_{-p}\bigl(\widehat{\nabla}_{p}^{\tau}K\bigr)\geq\widetilde {G}_{-p}(K), $$
(2.10)
with equality if and only if K is origin-symmetric.

3 Proofs of main results

In this section, we complete the proofs of Theorems 1.1-1.3. The proof of Theorem 1.1 requires the following lemma.

Lemma 3.1

If \(K, L \in S_{o}^{n}\) and \(p\geq1\), if \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), then for any \(Q\in S_{c}^{n}\)
$$ \frac{\widetilde{V}_{-p}(K, Q)}{V(K)}=\frac{\widetilde{V}_{-p}(L, Q)}{V(K)}. $$
(3.1)

Proof

Since \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), it easily follows that for any \(u\in S^{n-1}\)
$$h^{p}_{\Gamma_{p}^{+} K}(u)+h^{p}_{\Gamma_{p}^{-} K}(u)=h^{p}_{\Gamma_{p}^{+} L}(u)+h^{p}_{\Gamma_{p}^{-} L}(u). $$
Together (1.5) with (1.6), we get
$$\int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p} \biggl[ \frac{\rho _{K}^{n+p}(v)}{V(K)}+\frac{\rho_{-K}^{n+p}(v)}{V(-K)} -\frac{\rho_{L}^{n+p}(v)}{V(L)}-\frac{\rho_{-L}^{n+p}(v)}{V(-L)} \biggr]\,dv=0. $$
Let
$$\mu(v)=\frac{\rho_{K}^{n+p}(v)}{V(K)}+\frac{\rho_{-K}^{n+p}(v)}{V(-K)} -\frac{\rho_{L}^{n+p}(v)}{V(L)}- \frac{\rho_{-L}^{n+p}(v)}{V(-L)}, $$
then have
$$ \int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p} \mu(v)\,dv=0. $$
(3.2)
Notice that \(\rho_{-K}(v)=\rho_{K}(-v)\) for all \(v\in S^{n-1}\), thus we know that \(\mu(v)\) is a finite even Borel measure. Together with (3.2), then \(\mu(v)=0\), i.e.,
$$\frac{\rho_{K}^{n+p}(v)}{V(K)}+\frac{\rho_{K}^{n+p}(-v)}{V(-K)}= \frac{\rho_{L}^{n+p}(v)}{V(L)}+\frac{\rho_{L}^{n+p}(-v)}{V(L)}. $$
For any \(Q\in S_{c}^{n}\), then use \(\rho_{Q}(v)=\rho_{-Q}(v)=\rho_{Q}(-v)\) to get
$$\frac{\rho_{K}^{n+p}(v)\rho_{Q}^{-p}(v)}{V(K)}+\frac{\rho_{K}^{n+p} (-v)\rho_{Q}^{-p}(-v)}{V(K)}= \frac{\rho_{L}^{n+p}(v)\rho_{Q}^{-p}(v)}{V(L)}+\frac{\rho_{L}^{n+p} (-v)\rho_{Q}^{-p}(-v)}{V(L)}. $$
From (2.6), this yields for any \(Q\in S_{c}^{n}\)
$$\frac{\widetilde{V}_{-p}(K, Q)}{V(K)}=\frac{\widetilde{V}_{-p}(L, Q)}{V(L)}. $$
 □

Proof of Theorem 1.1

Together with definition (1.1), we know
$$ \frac{\omega_{n}^{-\frac{p}{n}}\widetilde{G}_{-p}(K)}{V(K)}= \inf \biggl\{ n\frac{\widetilde{V}_{-p}(K,Q)}{V(K)}V\bigl(Q^{\ast}\bigr)^{-\frac{p}{n}}: Q\in\mathcal{K}_{c}^{n} \biggr\} . $$
(3.3)
Since \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), from (3.1), we get, for any \(Q\in\mathcal{K}_{c}^{n}\),
$$ \frac{\widetilde{V}_{-p}(K, Q)}{V(K)}=\frac{\widetilde{V}_{-p}(L, Q)}{V(L)}. $$
(3.4)
Hence, from (3.3) and (3.4), we can get
$$\frac{\widetilde{G}_{-p}(K)}{V(K)}=\frac{\widetilde{G}_{-p}(L)}{V(L)}, $$
i.e.,
$$ \frac{\widetilde{G}_{-p}(K)}{\widetilde{G}_{-p}(L)}=\frac {V(K)}{V(L)}. $$
(3.5)
Taking \(Q=L\) in (3.4) and associating this with (2.8), since \(L\in\mathcal{K}_{c}^{n}\), we obtain
$$V(K)=\widetilde{V}_{-p}(K, L)\geq V(K)^{\frac{n+p}{n}}V(L)^{-\frac{p}{n}}, $$
i.e.,
$$ V(K)\leq V(L). $$
(3.6)
Combining (3.5) with (3.6), we get (1.9).

According to the equality condition of (3.6), we see that equality holds in (1.9) if and only if \(K=L\). □

Lemma 3.2

[44]

If \(K \in S_{o}^{n}\), \(p\geq1\), \(\tau \in(-1, 1)\), then
$$ \Gamma_{p}^{+}\bigl(\widehat{{\nabla}}^{\tau}_{p}K \bigr)=\Gamma_{p}^{\tau}K $$
(3.7)
and
$$ \Gamma_{p}^{-}\bigl(\widehat{{\nabla}}^{\tau}_{p}K \bigr)=\Gamma_{p}^{-\tau} K. $$
(3.8)

Proof of Theorem 1.2

Since L is not origin-symmetric and \(\tau \in(-1, 1)\), it follows from Theorem 2.B that \(\widetilde {G}_{-p}(\widehat{\nabla}_{p}^{\tau}L)> \widetilde{G}_{-p}(L)\). Choose \(\varepsilon>0\), such that \(K=(1-\varepsilon)\widehat{\nabla}_{p}^{\tau}L\) satisfies
$$\widetilde{G}_{-p}(K)=\widetilde{G}_{-p}\bigl((1- \varepsilon)\widehat{\nabla }_{p}^{\tau}L\bigr)> \widetilde{G}_{-p}(L). $$
By (3.7) and (3.8), we, respectively, have
$$\Gamma_{p}^{+} K=\Gamma_{p}^{+}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p}^{+} \bigl( \widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p}^{\tau}L \subset\Gamma_{p}^{\tau}L $$
and
$$\Gamma_{p}^{-} K=\Gamma_{p}^{-}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p}^{-} \bigl(\widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p}^{-\tau} L \subset\Gamma_{p}^{-\tau} L. $$
 □

Lemma 3.3

[44]

If \(K \in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
$$ \Gamma_{p}\bigl(\widehat{{\nabla}}^{\tau}_{p}K \bigr)=\Gamma_{p}K. $$
(3.9)

Proof of Theorem 1.3

Since L is not origin-symmetric, Theorem 2.B has \(\widetilde{G}_{-p}(\widehat{\nabla}_{p}^{\tau}L)> \widetilde {G}_{-p}(L)\) for \(\tau\in(-1, 1)\). Take \(\varepsilon>0\), and let \(K=(1-\varepsilon)\widehat{\nabla}_{p}^{\tau}L\) such that
$$\widetilde{G}_{-p}(K)=\widetilde{G}_{-p}\bigl((1- \varepsilon)\widehat{\nabla }_{p}^{\tau}L\bigr)> \widetilde{G}_{-p}(L). $$
It follows from (3.9) that
$$\Gamma_{p} K=\Gamma_{p}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p} \bigl(\widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p} L \subset\Gamma_{p} L. $$
 □

Declarations

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (Grant No. 11561020 and No. 11371224) and the Young Foundation of Hexi University (Grant No. QN2014-12). The referees of this paper proposed many very valuable comments and suggestions to improve the accuracy and readability of the original manuscript. We would like to express our most sincere thanks to the anonymous referees.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Hexi University

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