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 [6–10]).

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, 15–22]). 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, 24–32]).

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 [37–43]).

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

$$\Gamma_{p} K=\Gamma_{p}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p} \bigl(\widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p} L \subset\Gamma_{p} L. $$

 □