In this section, we prove Theorems 1.1-1.7. The proofs of Theorems 1.1 and 1.2 require the following lemma.
Lemma 3.1
If
\(K\in\mathcal{S}_{o}^{n}\)
and
\(\tau \in[-1,1]\), \(p>0\), then for
\(i< n-p\),
$$ C_{p,i}^{(\tau)}(K)\leq\widetilde{W}_{i}(K); $$
(3.1)
for
\(n-p< i< n\)
or
\(i>n\),
$$ C_{p,i}^{(\tau)}(K)\leq\widetilde{W}_{i}(K), $$
(3.2)
with equality if and only if
K
is centered. When
\(i=n\), (3.1) and (3.2) are identical.
Proof
From definitions (1.1) and (1.4), and using Minkowski’s inequality (see [32]), for \(i< n-p\), we have
$$\begin{aligned} C_{p,i}^{(\tau)}(K)^{\frac{p}{n-i}} =& \biggl[\frac{1}{n} \int_{\mathcal {S}^{n-1}}c_{p}^{(\tau)}(K,u)^{n-i}\,du \biggr]^{\frac{p}{n-i}} \\ =& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(f_{1}(\tau)\rho ^{p}(K,u)+f_{2}( \tau)\rho^{p}(-K,u) \bigr)^{\frac{n-i}{p}}\,du \biggr]^{\frac {p}{n-i}} \\ \leq& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(f_{1}(\tau)\rho ^{p}(K,u) \bigr)^{\frac{n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\ &{}+ \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(f_{2}(\tau)\rho ^{p}(-K,u) \bigr)^{\frac{n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\ =& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\rho^{p}(K,u) \bigr)^{\frac {n-i}{p}} \,du \biggr]^{\frac{p}{n-i}} \\ =& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}}\rho(K,u)^{n-i}\,du \biggr]^{\frac{p}{n-i}} \end{aligned}$$
i.e.
$$C_{p,i}^{(\tau)}(K)\leq\frac{1}{n} \int_{\mathcal{S}^{n-1}}\rho (K,u)^{n-i}\,du=\widetilde{W}_{i}(K). $$
Similarly, for \(n-p< i< n\) or \(i< n\), by Minkowski’s inequality we know (3.2) holds. From the equality condition of Minkowski’s inequality, we see that equalities hold in (3.1) and (3.2) if and only if K and −K are dilates of one another, that is, K is centered. □
In Lemma 3.1, let \(i=0\) and notice (2.2), we have the following corollary immediately.
Corollary 3.1
If
\(K\in\mathcal{S}_{o}^{n}\)
and
\(\tau\in[-1,1]\), \(p>0\), then for
\(p< n\),
$$ C_{p}^{(\tau)}(K)\leq V(K); $$
(3.3)
for
\(p>n\),
$$ C_{p}^{(\tau)}(K)\geq V(K), $$
(3.4)
with equality if and only if
K
is centered.
Theorem 2.A leads to the following corollary.
Corollary 3.2
If
\(K\in\mathcal{S}_{os}^{n}\)
and
\(\tau\in[-1,1]\), \(p>0\), then for
\(p< n\)
and
\(n\geq1\),
$$ C_{p}^{(\tau)}(K)C_{p}^{(\tau)} \bigl(K^{\ast}\bigr)\leq\omega_{n}^{2}, $$
(3.5)
with equality in each inequality if and only if
K
is an ellipsoid centered at the origin.
Proof of Theorem 1.1
Using Hölder’s inequality (see [32]), we have for \(p< n\) and \(n\geq1\),
$$\begin{aligned} C_{p}^{(\tau)}(K_{1},\ldots, K_{n}) =& \biggl[\frac{1}{n} \int_{\mathcal {S}^{n-1}}c_{p}^{(\tau)}(K_{1}, u) \cdots c_{p}^{(\tau)}(K_{n}, u)\,du \biggr]^{n\cdot\frac{1}{n}} \\ \leq& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(c_{p}^{(\tau )}(K_{1}, u) \bigr)^{n}\,du \biggr]^{\frac{1}{n}}\times\cdots\times \biggl[ \frac {1}{n} \int_{\mathcal{S}^{n-1}} \bigl(c_{p}^{(\tau)}(K_{n}, u) \bigr)^{n}\,du \biggr]^{\frac{1}{n}} \\ =&C_{p}^{(\tau)}(K_{1})^{\frac{1}{n}}\cdots C_{p}^{(\tau)}(K_{n})^{\frac {1}{n}} \\ \leq& V(K_{1})^{\frac{1}{n}}\cdots V(K_{n})^{\frac{1}{n}} \end{aligned}$$
i.e.
\(C_{p}^{(\tau)}(K_{1},\ldots, K_{n})^{n}\leq V(K_{1})\cdots V(K_{n})\).
Similarly, we can obtain \(C_{p}^{(\tau)}(K_{1},\ldots, K_{n})^{n}\geq V(K_{1})\cdots V(K_{n})\) when \(p>n\). The equality conditions of (3.3) and (3.4) tell us that equalities hold in Theorem 1.1 if and only if \(K_{1},\ldots, K_{n}\) are dilates of each other. □
Proof of Theorem 1.2
From the proof of Lemma 3.1, for \(i< n-p\), we get
$$ C_{p,i}^{(\tau)}(K)\leq\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\rho ^{p}(K,u) \bigr)^{\frac{n-i}{p}} \,du \quad \mbox{or}\quad \frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\rho^{p}(-K,u) \bigr)^{\frac{n-i}{p}} \,du. $$
(3.6)
Notice (1.2), \(f_{1}(+1)=1\), \(f_{2}(+1)=0\), we have
$$C_{p,i}^{(+1)}(K)=\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\rho ^{p}(K,u) \bigr)^{\frac{n-i}{p}} \,du. $$
Similarly, we obtain
$$C_{p,i}^{(-1)}(K)=\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\rho ^{p}(-K,u) \bigr)^{\frac{n-i}{p}}\,du. $$
Together with (3.6), it follows that
$$C_{p,i}^{(\tau)}(K)\leq C_{p,i}^{(\pm1)}(K), $$
which is just the right inequality of (1.7).
According the equality condition of Lemma 3.1, we know that if \(\tau\neq \pm1\), equality holds in the right inequality of (1.7) if and only if K is centered, that means that, if K is not origin-symmetric, equality holds in the right inequality if and only if \(\tau=\pm1\).
Now we prove the left inequality of (1.7). When \(\tau=0\), we have \(f_{1}(\tau)=f_{2}(\tau)=\frac{1}{2}\), thus, by Minkowski’s inequality,
$$\begin{aligned} C_{p,i}^{(0)}(K)^{\frac{p}{n-i}} =& \biggl[\frac{1}{n} \int_{\mathcal {S}^{n-1}} \biggl(\frac{1}{2}\rho^{p}(K,u)+ \frac{1}{2}\rho^{p}(-K,u) \biggr)^{\frac{n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\ =& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \biggl(\frac{1}{2}f_{1}(\tau ) \rho^{p}(K,u)+\frac{1}{2}f_{2}(\tau) \rho^{p}(-K,u) \\ &\biggl.\biggl.\biggl.\biggl.{}+\frac{1}{2}f_{1}(\tau)\rho^{p}(-K,u)+ \frac{1}{2}f_{2}(\tau )\rho^{p}(K,u) \biggr)\biggr.^{\frac{n-i}{p}}\,du \biggr]\biggr.^{\frac{p}{n-i}} \\ \leq& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \biggl(\frac{1}{2}f_{1}(\tau ) \rho^{p}(K,u)+\frac{1}{2}f_{2}(\tau) \rho^{p}(-K,u) \biggr)^{\frac {n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\ &{}+ \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \biggl(\frac {1}{2}f_{1}(\tau) \rho^{p}(-K,u)+\frac{1}{2}f_{2}(\tau) \rho^{p}(K,u) \biggr)^{\frac{n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\ =& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(f_{1}(\tau)\rho ^{p}(K,u)+f_{2}( \tau)\rho^{p}(-K,u) \bigr)^{\frac{n-i}{p}}\,du \biggr]^{\frac {p}{n-i}} \\ =& \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau )}(K,u)^{n-i}\,du \biggr]^{\frac{p}{n-i}} \\ =&C_{p,i}^{(\tau)}(K)^{\frac{p}{n-i}} \end{aligned}$$
i.e.
\(C_{p,i}^{(0)}(K)\leq C_{p,i}^{(\tau)}(K)\), which just is the left inequality of (1.7).
The equality condition of Minkowski’s inequality tells us that if \(\tau \neq0\), equality holds in the left inequality of (1.7) if and only if K is centered, then if K is not origin-symmetric, equality holds in the right inequality if and only if \(\tau=0\).
For the case \(n-p< i< n\) or \(i>n\), the proof is similar. □
The proof of Theorem 1.3 requires the following inequality (see [29]).
Lemma 3.2
If
\(f_{0}, f_{1}, \ldots,f_{m}\)
are strictly positive continuous functions defined on
\(\mathcal{S}^{n-1}\)
and
\(\lambda_{1}, \ldots, \lambda_{m}\)
are positive constants the sum of whose reciprocals is unity, then
$$ \int_{\mathcal{S}^{n-1}}f_{0}(u)f_{1}(u)\cdots f_{m}(u)\,du\leq\prod^{m}_{i=1} \biggl[ \int_{\mathcal{S}^{n-1}}f_{0}(u)f_{i}^{\lambda _{i}}(u) \,du\biggr]^{\frac{1}{\lambda_{i}}}, $$
(3.7)
with equality if and only if there exist positive constants
\(\alpha _{1}, \ldots, \alpha_{m}\)
such that
\(\alpha_{1}f_{1}^{\lambda _{1}}(u)=\cdots=\alpha_{m}f_{m}^{\lambda_{m}}(u)\)
for all
\(u\in S^{n-1}\).
Proof of Theorem 1.3
For \(K_{1},\ldots, K_{n} \in S_{o}^{n}\), in (3.7), take \(\lambda_{1}=\lambda_{2}=\cdots=\lambda_{m}\), and
$$\begin{aligned}& f_{0}=c_{p}^{(\tau)}(K_{1},u)\cdots c_{p}^{(\tau )}(K_{n-m},u) \quad (f_{0}=1\mbox{ if } m=n), \\& f_{i}=c_{p}^{(\tau)}(K_{n-i+1},u) \quad (1\leq i\leq m), \end{aligned}$$
then it follows that
$$\begin{aligned}& \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau)}(K_{1},u) \cdots c_{p}^{(\tau)}(K_{n},u)\,du \\& \quad \leq\prod ^{m}_{i=1}\biggl[ \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau )}(K_{1},u) \cdots c_{p}^{(\tau)}(K_{n-m},u)c_{p}^{(\tau )}(K_{n-i+1},u)^{m} \,du\biggr]^{\frac{1}{m}}. \end{aligned}$$
By definition (1.1), this yields
$$C_{p}^{(\tau)}(K_{1},\ldots, K_{n})^{m} \leq\prod^{m}_{i=1}C_{p}^{(\tau )}(K_{1}, \ldots, K_{n-m},K_{n-i+1},K_{n-i+1},\ldots, K_{n-i+1}). $$
From the equality condition of inequality (3.7), we can see that equality holds in Theorem 1.3 if and only if \(K_{n-m+1},\ldots, K_{n}\) all have a similar general \(L_{p}\)-chord. □
When \(m=n\) in Theorem 1.3 and use (1.4), we have the following corollary.
Corollary 3.3
If
\(K_{1},\ldots, K_{n} \in\mathcal {S}_{o}^{n}\)
and
\(\tau\in[-1,1]\), \(p>0\), then
$$C_{p}^{(\tau)}(K_{1},\ldots, K_{n})^{n} \leq C_{p}^{(\tau )}(K_{1})C_{p}^{(\tau)}(K_{2}) \cdots C_{p}^{(\tau)}(K_{n}), $$
with equality if and only if
\(K_{1},\ldots, K_{n}\)
all have a similar general
\(L_{p}\)-chord.
Proof of Theorem 1.4
For \(i< j <k\), by Hölder’s inequality (see [32]), we have
$$\begin{aligned}& C_{p,i}^{(\tau)}(K,L)^{\frac{{k-j}}{k-i}}C_{p,k}^{(\tau)}(K,L)^{\frac {{j-i}}{k-i}} \\& \quad = \biggl(\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau )}(K,u)^{n-i}c_{p}^{(\tau)}(L,u)^{i} \,du \biggr)^{\frac{{k-j}}{k-i}} \biggl(\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau )}(K,u)^{n-k}c_{p}^{(\tau)}(L,u)^{k} \,du \biggr)^{\frac{{j-i}}{k-i}} \\& \quad \geq\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau)}(K,u)^{\frac {(n-i)(k-j)+(n-k)(j-i)}{k-i}}c_{p}^{(\tau)}(L,u)^{\frac {i(k-j)+k(j-i)}{k-i}} \,du \\& \quad =\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau)}(K)^{n-j}c_{p}^{(\tau )}(L,u)^{j} \,du \\& \quad =C_{p,j}^{(\tau)}(K,L), \end{aligned}$$
that is,
$$C_{p,j}^{(\tau)}(K,L)^{k-i}\leq C_{p,i}^{(\tau )}(K,L)^{k-j}C_{p,k}^{(\tau)}(K,L)^{j-i}. $$
From the equality condition of Hölder’s inequality, we know that equality holds in (1.10) if and only if K and L have a similar general \(L_{p}\)-chord. □
Letting \(i=0\), \(k=n\) in Theorem 1.4, we have the following fact.
Corollary 3.4
If
\(K,L \in\mathcal{S}_{o}^{n}\)
and
\(\tau\in[-1,1]\), \(p>0\), then for
\(0\leq j \leq n\),
$$ C_{p,j}^{(\tau)}(K,L)^{n}\leq C_{p}^{(\tau)}(K)^{n-j}C_{p}^{(\tau )}(L)^{j}. $$
(3.8)
For
\(j< 0\)
or
\(j> n\), the inequality (3.8) is reversed, with equality if and only if
K
and
L
have a similar general
\(L_{p}\)-chord.
Let \(j=1\) in Corollary 3.4, we obtain the following dual Minkowski inequality for the general \(L_{p}\)-mixed chord integrals.
Corollary 3.5
If
\(K,L \in\mathcal{S}_{o}^{n}\)
and
\(\tau\in[-1,1]\), \(p>0\), then
$$ C_{p,1}^{(\tau)}(K,L)^{n}\leq C_{p}^{(\tau)}(K)^{n-1}C_{p}^{(\tau)}(L), $$
(3.9)
with equality if and only if
K
and
L
have a similar general
\(L_{p}\)-chord.
Now we prove the associated inequalities based on \(L_{p}\)-radial combinations.
Proof of Theorem 1.5
For \(i\leq n-p\), it follows from Minkowski’s inequality (see [32]) that
$$\begin{aligned}& C_{p,i}^{(\tau)}(\lambda\circ K \, \tilde{+}_{p}\, \mu \circ L)^{\frac {p}{n-i}} \\& \quad = \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(\tau)}(\lambda\circ K \, \tilde{+}_{p}\, \mu\circ L,u)^{n-i}\,du \biggr]^{\frac{p}{n-i}} \\& \quad = \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(f_{1}(\tau)\rho ^{p}( \lambda\circ K \, \tilde{+}_{p}\, \mu\circ L,u)+f_{2}(\tau) \rho ^{p}(\lambda\circ K \, \tilde{+}_{p}\, \mu\circ L,-u) \bigr)^{\frac {n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\& \quad = \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\lambda f_{1}(\tau)\rho ^{p}(K,u)+\lambda f_{2}(\tau)\rho^{p}(-K,u) \\& \qquad\biggl.\biggl.\bigl.\bigl.{}+\mu f_{1}(\tau)\rho^{p}(L,u)+\mu f_{2}(\tau)\rho ^{p}(-L,u) \bigr)\bigr.^{\frac{n-i}{p}}\,du \biggr]\biggr.^{\frac{p}{n-i}} \\& \quad \leq \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\lambda f_{1}(\tau ) \rho^{p}(K,u)+\lambda f_{2}(\tau)\rho^{p}(-K,u) \bigr)^{\frac {n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\& \qquad {}+ \biggl[\frac{1}{n} \int_{\mathcal{S}^{n-1}} \bigl(\mu f_{1}(\tau ) \rho^{p}(L,u)+\mu f_{2}(\tau)\rho^{p}(-L,u) \bigr)^{\frac{n-i}{p}}\,du \biggr]^{\frac{p}{n-i}} \\& \quad =\lambda C_{p,i}^{(\tau)}(K)^{\frac{1}{n-i}}+\mu C_{p,i}^{(\tau )}(L)^{\frac{1}{n-i}}. \end{aligned}$$
Similarly, we can prove the case of \(n-p< i< n\) or \(i>n\). From the equality conditions of Minkowski’s inequality, we know that equality holds in Theorem 1.5 if and only if K and L have a similar general \(L_{p}\)-chord. □
In order to prove Theorem 1.6, we require the following lemma which is an extended form of Beckenbach’s inequality (see [33]) obtained by Dresher (see [34]) through the means of moment-space techniques.
Lemma 3.3
(The Beckenbach-Dresher inequality)
If
\(p\geq1\geq r\geq0\), \(f, g\geq0\), and
ϕ
is a distribution function, then
$$ \biggl(\frac{\int_{E}(f+g)^{p}\,d\phi}{\int_{E}(f+g)^{r}\,d\phi} \biggr)^{\frac {1}{p-r}}\leq \biggl(\frac{\int_{E}f^{p}\,d\phi}{\int_{E}f^{r}\,d\phi} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\int_{E}g^{p}\,d\phi}{\int_{E}g^{r}\,d\phi} \biggr)^{\frac{1}{p-r}}, $$
(3.10)
with equality if and only if the functions
f
and
g
are positively proportional.
Moreover, the inverse Beckenbach-Dresher inequality was obtained by Li and Zhao (see [35]).
Lemma 3.4
(The Inverse Beckenbach-Dresher inequality)
If
\(r\leq0\leq p\leq1\), \(f, g\geq0\), and
ϕ
is a distribution function, then
$$ \biggl(\frac{\int_{E}(f+g)^{p}\,d\phi}{\int_{E}(f+g)^{r}\,d\phi} \biggr)^{\frac {1}{p-r}}\geq \biggl(\frac{\int_{E}f^{p}\,d\phi}{\int_{E}f^{r}\,d\phi} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\int_{E}g^{p}\,d\phi}{\int_{E}g^{r}\,d\phi} \biggr)^{\frac{1}{p-r}}, $$
(3.11)
with equality if and only if the functions
f
and
g
are positively proportional.
Proof of Theorem 1.6
By definitions (1.4) and (2.4), we have
$$\begin{aligned}& \biggl( \frac{C_{p,i}^{(\tau)}(K\, \tilde{+}_{p}\, L)}{C_{p,j}^{(\tau )}(K\, \tilde{+}_{p}\, L)} \biggr)^{\frac{p}{j-i}} \\& \quad = \biggl( \frac{\frac{1}{n}\int_{\mathcal{S}^{n-1}}c_{p}^{(\tau)}(K\, \tilde {+}_{p}\, L,u)^{n-i}\,du}{\frac{1}{n}\int_{\mathcal{S}^{n-1}}c_{p}^{(\tau )}(K\, \tilde{+}_{p}\, L,u)^{n-j}\,du} \biggr)^{\frac{p}{j-i}} \\& \quad = \biggl( \frac{\frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho^{p}(K \, \tilde{+}_{p}\, L,u)+f_{2}(\tau)\rho^{p}(K \, \tilde{+}_{p}\, L,-u)]^{\frac{{n-i}}{p}}\,du}{ \frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho^{p}(K \, \tilde {+}_{p}\, L,u)+f_{2}(\tau)\rho^{p}(K \, \tilde{+}_{p}\, L,-u)]^{\frac {{n-j}}{p}}\,du} \biggr)^{\frac{p}{j-i}} \\& \quad = \biggl( \frac{\frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho ^{p}(K,u)+f_{2}(\tau)\rho^{p}(-K,u)+f_{1}(\tau)\rho^{p}(L,u)+f_{2}(\tau )\rho^{p}(-L,u)]^{\frac{{n-i}}{p}}\,du}{ \frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho^{p}(K,u)+f_{2}(\tau )\rho^{p}(-K,u)+f_{1}(\tau)\rho^{p}(L,u)+f_{2}(\tau)\rho ^{p}(-L,u)]^{\frac{{n-j}}{p}}\,du} \biggr)^{\frac{p}{j-i}} \\& \quad \leq \biggl( \frac{\frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho ^{p}(K,u)+f_{2}(\tau)\rho^{p}(-K,u)]^{\frac{{n-i}}{p}}\,du}{ \frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho^{p}(K,u)+f_{2}(\tau )\rho^{p}(-K,u)]^{\frac{{n-j}}{p}}\,du} \biggr)^{\frac{p}{j-i}} \\& \qquad {}+ \biggl(\frac{\frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho ^{p}(L,u)+f_{2}(\tau)\rho^{p}(-L,u)]^{\frac{{n-i}}{p}}\,du}{ \frac{1}{n}\int_{\mathcal{S}^{n-1}}[f_{1}(\tau)\rho^{p}(L,u)+f_{2}(\tau )\rho^{p}(-L,u)]^{\frac{{n-j}}{p}}\,du} \biggr)^{\frac{p}{j-i}} \\& \quad = \biggl( \frac{C_{p,i}^{(\tau)}(K)}{C_{p,j}^{(\tau)}(K)} \biggr)^{\frac {p}{j-i}}+ \biggl( \frac{C_{p,i}^{(\tau)}(L)}{C_{p,j}^{(\tau)}(L)} \biggr)^{\frac{p}{j-i}}, \end{aligned}$$
inequality (1.13) is proved. Using the same method, we can see that when \(j\geq n\geq i\geq n-p\), the reversed inequality of (1.13) is obtained.
From the equality condition of Lemma 3.3 and Lemma 3.4, we know that equality in inequality (1.13) if and only if K and L have a similar general \(L_{p}\)-chord. □
Finally, we prove the interesting equality for the \(L_{p}\)-radial bodies.
Proof of Theorem 1.7
For \(\forall\tau\in[-1,1]\),
$$\begin{aligned} C_{p,i}^{(\tau)}(\bar {\vartriangle}_{p}K) =& \frac{1}{n} \int _{\mathcal{S}^{n-1}}c_{p}^{(\tau)}(\bar {\vartriangle }_{p}K,u)^{n-i}\,du \\ =&\frac{1}{n} \int_{\mathcal{S}^{n-1}}\bigl[f_{1}(\tau)\rho^{p}( \bar {\vartriangle}_{p}K,u)+f_{2}(\tau) \rho^{p}(\bar {\vartriangle }_{p}-K,u) \bigr]^{\frac{{n-i}}{p}}\,du \\ =&\frac{1}{n} \int_{\mathcal{S}^{n-1}}\biggl[\frac{1}{2}f_{1}(\tau)\rho ^{p}(K,u)+\frac{1}{2}f_{2}(\tau)\rho^{p}(-K,u) \\ &\biggl.\biggl.{}+\frac{1}{2}f_{1}(\tau)\rho^{p}(-K,u)+ \frac{1}{2}f_{2}(\tau)\rho ^{p}(K,u) \biggr]\biggr.^{\frac{{n-i}}{p}}\,du \\ =&\frac{1}{n} \int_{\mathcal{S}^{n-1}}\biggl[\frac{1}{2}\rho^{p}(K,u)+ \frac {1}{2}\rho^{p}(-K,u)\biggr]^{\frac{{n-i}}{p}}\,du \\ =&\frac{1}{n} \int_{\mathcal{S}^{n-1}}c_{p}^{(0)}(K,u)^{n-i} \,du \\ =&C_{p,i}(K), \end{aligned}$$
that is, \(C_{p,i}^{(\tau)}(\bar {\vartriangle}_{p}K)=C_{p,i}(K)\). □