The support function \(h_{K}=h(K,\cdot): S^{n-1}\rightarrow\mathbb{R}\) of a compact convex set \(K\subset\mathbb{R}^{n}\) is defined, for \(u\in S^{n-1}\), by
$$h_{K}(u)=\max\{u\cdot x: x\in K\}, $$
and it uniquely determines the compact convex set. Here \(u\cdot x\) denotes the standard inner product of u and x in \(\mathbb{R}^{n}\).
For \(K, L\in\mathcal{K}^{n}\), and \(\lambda,\mu\geq0\) (not both zero), the Minkowski linear combination \(\lambda\cdot K+ \mu\cdot L\in \mathcal{K}^{n}\) is defined by
$$h(\lambda\cdot K+ \mu\cdot L, \cdot)=\lambda h(K,\cdot)+\mu h(L,\cdot). $$
The classical Brunn-Minkowski inequality (see [24]) states that for convex bodies \(K,L\in\mathcal{K}^{n}\) and real \(\lambda,\mu \geq0\) (not both zero), the volume of the bodies and of their Minkowski linear combination \(\lambda\cdot K+ \mu\cdot L\in\mathcal{K}^{n}\) are related by
$$ V(\lambda\cdot K+ \mu\cdot L)^{\frac{1}{n}}\geq\lambda V(K)^{\frac {1}{n}}+\mu V(L)^{\frac{1}{n}}, $$
(2.1)
with equality if and only if K and L are homothetic.
For real \(p\geq1\), \(K,L\in\mathcal{K}^{n}_{o}\), and \(\lambda,\mu\geq0\) (not both zero), the Firey linear combination \(\lambda\cdot K+_{p}\mu\cdot L\), is defined by (see [25])
$$h(\lambda\cdot K+ _{p}\mu\cdot L, \cdot)^{p}=\lambda h(K, \cdot )^{p}+\mu h(L,\cdot)^{p}. $$
Firey [25] also established the following \(L_{p}\) Brunn-Minkowski inequality. If \(p>1\), \(\lambda,\mu\geq0\) (not both zero), and \(K,L\in\mathcal {K}^{n}_{o}\), then
$$V(\lambda\cdot K+_{p} \mu\cdot L)^{\frac{p}{n}}\geq \lambda V(K)^{\frac{p}{n}}+\mu V(L)^{\frac{p}{n}}, $$
with equality if and only if K and L are dilates.
The radial function of \(K, \rho_{K}: S^{n-1}\rightarrow[0,\infty)\), is defined by
$$\rho_{K}(u)=\max\{\lambda: \lambda u\in K\}. $$
A set \(K\subset\mathbb{R}^{n}\) is said to be a star body about the origin, if the line segment from the origin to any point \(x\in K\) is contained in K and K has continuous and positive radial function \(\rho _{K}(\cdot)\).
Note that \(K\in\mathcal{S}^{n}_{o}\) can be uniquely determined by its radial function \(\rho_{K}(\cdot)\) and vice versa. If \(\lambda>0\), we have
$$\rho_{K}(\lambda x)=\lambda^{-1} \rho_{K}(x)\quad \mbox{and} \quad\rho _{\lambda K}(x)=\lambda \rho_{K}(x). $$
More generally, from the definition of the radial function it follows immediately that for \(\Lambda\in\operatorname{GL}(n)\) the radial function of the image \(\Lambda K=\{\Lambda y: y\in K\}\) of \(K\in\mathcal {S}^{n}_{o}\) is given by (see [26])
$$ \rho(\Lambda K,x)=\rho\bigl(K,\Lambda^{-1}x\bigr), \quad\mbox{for all } x\in \mathbb{R}^{n}. $$
(2.2)
Two star bodies \(K,L\in\mathcal{S}^{n}_{o}\) are said to be dilates of each other if there is a constant \(\lambda>0\) such that \(L=\lambda K\), and the equation \(\rho_{L}(u)=\lambda\rho_{K}(u)\) for all \(u\in S^{n-1}\). Clearly, for \(K,L\in\mathcal{S}^{n}_{o}\),
$$K\subseteq L \quad\mbox{if and only if}\quad \rho_{K}(u)\leq\rho _{L}(u),\quad \mbox{for all } u\in S^{n-1}. $$
The natural metric on \(\mathcal{S}^{n}_{o}\) is the radial metric \(\widetilde{\delta}(\cdot,\cdot): \mathcal{S}^{n}_{o}\times\mathcal {S}^{n}_{o}\rightarrow\mathbb{R}\) defined as
$$\widetilde{\delta}(K,L)=\|\rho_{K}-\rho_{L}\|_{\infty}= \sup_{u\in S^{n-1}}\bigl|\rho_{K}(u)-\rho_{L}(u)\bigr|, \quad\mbox{for } K,L\in\mathcal{S}^{n}_{o}. $$
A sequence of star bodies \(\{K_{j}\}\subset\mathcal{S}^{n}_{o}\) is said to be convergent to \(K\in\mathcal{S}^{n}_{o}\) in δ̃ if \(\widetilde{\delta}(K_{j},K)\rightarrow0\) as \(j\rightarrow\infty\), and equivalently, \(\rho_{K_{j}}\) is uniformly convergent to \(\rho_{K}\) on \(S^{n-1}\).
If \(K\in\mathcal{K}_{o}^{n}\), the polar body, \(K^{\ast}\), of K is defined by
$$\begin{aligned} K^{*}=\bigl\{ x\in\mathbb{R}^{n}: x\cdot y \leq1, \mbox{ for all } y\in K\bigr\} . \end{aligned}$$
Obviously, we have \((K^{*})^{*}=K\).
The Blaschke-Santaló inequality [27] is one of the fundamental affine isoperimetric inequalities. It states that if \(Q\in \mathcal{K}^{n}_{c}\) then
$$ V(Q)V\bigl(Q^{*}\bigr)\leq\omega_{n}^{2}, $$
(2.3)
with equality if and only if Q is an ellipsoid.
For \(\phi\in\Phi\) and \(\lambda, \mu\geq0\) (not both zero), we define the Orlicz radial sum \(\lambda\circ K\widetilde{+}_{-\phi}\mu\circ L\) of two star bodies \(K, L\in\mathcal{S}^{n}_{o}\), by
$$ \rho(\lambda\circ K\widetilde{+}_{-\phi}\mu\circ L,u)=\sup \biggl\{ t>0: \lambda\phi \biggl(\frac{t}{\rho_{K}(u)} \biggr)+\mu\phi \biggl( \frac {t}{\rho_{L}(u)} \biggr)\leq1 \biggr\} , $$
(2.4)
for all \(u\in S^{n-1}\).
When \(\phi(t)=t^{p}\), with \(p\geq1\), it is easy to show that the Orlicz radial sum reduces to Lutwak’s p-harmonic radial combination (see [6]):
$$\rho(\mu\circ K\widetilde{+}_{-p}\mu\circ L, \cdot)^{-p}= \lambda\rho (K,\cdot)^{-p}+\mu\rho(L,\cdot)^{-p}. $$
If \(K,L\in\mathcal{K}^{n}_{o}\), then
$$\lambda\circ K\widetilde{+}_{-p} \mu\circ L=\bigl(\lambda\cdot K^{*}+_{p}\mu\cdot L^{*}\bigr)^{*}. $$
We denote the right derivative of a real-valued function f by \(f_{r}'\). For \(\phi\in\Phi\), there is \(\phi_{r}'(1)>0\) because ϕ is convex and strictly increasing.
Let \(\phi\in\Phi\). By the Orlicz radial sum (2.4), we define the dual Orlicz mixed volume \(\widetilde{V}_{-\phi}(K,L)\) of convex bodies \(K,L\in\mathcal{S}^{n}_{o}\) by
$$ \frac{n}{-\phi'_{r}(1)}\widetilde{V}_{-\phi}(K,L)=\lim _{\varepsilon \rightarrow0^{+}}\frac{V(K\widetilde{+}_{-\phi}\varepsilon\circ L)-V(K)}{\varepsilon}. $$
(2.5)
From (2.5) we easily obtain the following integral formula of the dual Orlicz mixed volume:
$$ \widetilde{V}_{-\phi}(K,L)=\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho _{K}(u)}{\rho_{L}(u)} \biggr) \rho_{K}^{n}(u)\,\mathrm{d}S(u). $$
(2.6)
Apparently, we have
$$ \widetilde{V}_{-\phi}(K,K)=V(K). $$
(2.7)
For \(\phi(t)=t^{p}\) with \(p\geq1\), the dual Orlicz mixed volume \(\widetilde{V}_{\phi}(K,L)\) reduces to Lutwak’s p-dual mixed volume formula (see [6]):
$$\widetilde{V}_{-p}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho_{K}^{n+p}(u)\rho _{L}^{-p}(u)\,\mathrm{d}S(u), $$
for all \(K,L\in\mathcal{S}^{n}_{o}\).
Using the same argument as in [23] we establish the following dual Orlicz-Minkowski inequality: Suppose \(\phi\in\Phi\). If \(K,L\in \mathcal{S}^{n}_{o}\), then
$$ \widetilde{V}_{-\phi}(K,L)\geq V(K)\phi \biggl( \biggl( \frac {V(K)}{V(L)} \biggr)^{\frac{1}{n}} \biggr), $$
(2.8)
with equality if and only if K and L are dilates of each other.
When \(\phi(t)=t^{p}\), with \(p\geq1\). Then (2.8) reduces to the following \(L_{p}\)-dual Minkowski inequality (see [6]):
$$\widetilde{V}_{-p}(K,L)^{n}\geq V(K)^{n+p}V(L)^{-p}, $$
with equality if and only if K and L are dilates of each other.
We also establish the dual Orlicz-Brunn-Minkowski inequality as follows: Suppose \(K,L\in\mathcal{S}^{n}_{o}\), and \(\lambda,\mu>0\). If \(\phi\in\Phi\), then
$$\lambda\phi \biggl( \biggl(\frac{V(\lambda\circ K\widetilde{+}_{\phi}\mu \circ L)}{V(K)} \biggr)^{\frac{1}{n}} \biggr)+\mu\phi \biggl( \biggl(\frac {V(\lambda\circ K\widetilde{+}_{\phi}\mu\circ L)}{V(L)} \biggr)^{\frac {1}{n}} \biggr)\leq1, $$
with equality if and only if K and L are dilates of each other.
When \(\phi(t)=t^{p}\), with \(p\geq1\), the above inequality reduces to Lutwak’s \(L_{p}\)-dual Brunn-Minkowski inequality (see [6]):
$$V(\lambda\circ K\widetilde{+}_{-p}\mu\circ L)^{-\frac{p}{n}}\geq \lambda V(K)^{-\frac{p}{n}}+\mu V(L)^{-\frac{p}{n}}, $$
with equality if and only if K and L are dilates of each other.
The following results are required in the proofs of our main results.
Lemma 2.1
If
\(\phi\in\Phi\), and
\(K,L\in\mathcal{S}^{n}_{o}\), then for
\(\Lambda \in\operatorname{SL}(n)\),
$$\widetilde{V}_{-\phi}(\Lambda K,\Lambda L)=\widetilde{V}_{\phi}(K,L). $$
Proof
For \(x\in\mathbb{R}^{n}\backslash\{0\}\), let \(\langle x\rangle=x/|x|\). By (2.6) and (2.2), we have
$$\begin{aligned} \widetilde{V}_{-\phi}(\Lambda K,L) =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{\Lambda K}(u)}{\rho_{L}(u)} \biggr) \rho_{\Lambda K}^{n}(u)\,\mathrm{d}S(u)\\ =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(\Lambda^{-1} u)}{\rho_{L}(\Lambda\Lambda^{-1}u)} \biggr) \rho_{K}^{n}\bigl(\Lambda ^{-1}u\bigr)\,\mathrm{d}S \bigl(\Lambda\Lambda^{-1}u\bigr)\\ =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(\langle \Lambda ^{-1} u\rangle)}{\rho_{\Lambda^{-1} L}(\langle\Lambda^{-1}u\rangle )} \biggr) \rho_{K}^{n}\bigl(\bigl\langle \Lambda^{-1}u\bigr\rangle \bigr)|\det\Lambda|\,\mathrm{d}S\bigl(\bigl\langle \Lambda^{-1}u\bigr\rangle \bigr)\\ =&\widetilde{V}_{-\phi}\bigl(K,\Lambda^{-1}L\bigr), \end{aligned}$$
where \(\Lambda^{-1}\) denotes the inverse of Λ. □
It is easy to check that \(\widetilde{V}_{-\phi}(\lambda K,\lambda L)=\lambda^{n}\widetilde{V}_{-\phi}(K,L)\), for \(\lambda>0\). Therefore, we have the following.
Proposition 2.2
Suppose
\(K, L\in\mathcal{S}^{n}_{o}\). If
\(\phi\in\Phi\)
and
\(\Lambda \in\operatorname{GL}(n)\), then
$$\widetilde{V}_{-\phi}(\Lambda K,\Lambda L)=|\det\Lambda|\widetilde {V}_{-\phi}(K,L). $$
Lemma 2.3
Suppose
\(f_{i}\), f
are strictly positive and continuous functions on
\(S^{n-1}\); \(\phi_{j},\phi\in\Phi\); \(\mu_{k}\), μ
are Borel probability measures on
\(S^{n-1}\); \(i,j,k\in\mathbb{N}\). If
\(f_{i}\rightarrow f\)
pointwise, \(\phi _{j}\rightarrow\phi\)
uniformly, and
\(\mu_{k}\rightarrow\mu\)
weakly, then
$$ \int_{S^{n-1}}\phi_{j}(f_{i})\,\mathrm{d} \mu_{k}\rightarrow \int _{S^{n-1}}\phi(f)\,\mathrm{d}\mu, \quad\textit{as } i,j,k\rightarrow \infty. $$
(2.9)
Proof
The continuity of \(f_{i}\) and f, and \(f_{i}\rightarrow f\) pointwise guarantee that \(f_{i}\rightarrow f\) uniformly. Thus, there exists an \(N_{0}\in\mathbb{N}\), such that
$$\frac{1}{2}\min_{u\in S^{n-1}}f(u)\leq f_{i}(u)\leq2 \max_{u\in S^{n-1}}f(u), \quad\mbox{for } i>N_{0}. $$
Let
$$c_{m}=\min \biggl\{ \frac{1}{2}\min_{u\in S^{n-1}}f(u), \min_{u\in S^{n-1}}f_{i}(u), \mbox{ with } i\leq N_{0} \biggr\} $$
and
$$c_{M}=\min \Bigl\{ 2\max_{u\in S^{n-1}}f(u),\max _{u\in S^{n-1}}f_{i}(u), \mbox{ with } i\leq N_{0} \Bigr\} . $$
The strict positivity and the continuity of \(f_{i}\) and f imply that \(0< c_{m}\leq c_{M}<\infty\). Thus,
$$ c_{m}\leq f(u)\leq c_{M}\quad \mbox{and}\quad c_{m}\leq f_{i}(u)\leq c_{M}, \quad\mbox{for } u\in S^{n-1} \mbox{ and } i\in\mathbb{N}. $$
(2.10)
Since \(\phi_{j}\rightarrow\phi\) uniformly on \([c_{m}, c_{M}]\), by (2.10) and \(f_{i}\rightarrow f\) uniformly, it follows that as \(i,j\rightarrow\infty\), \(\phi_{j}(f_{i})\rightarrow\phi(f)\), uniformly on \(S^{n-1}\). Combined with \(\mu_{k}\rightarrow\mu\) weakly, one concludes that, as \(i,j,k\rightarrow\infty\),
$$\int_{S^{n-1}}\phi_{j}(f_{i})\,\mathrm{d} \mu_{k}\rightarrow \int _{S^{n-1}}\phi(f)\,\mathrm{d}\mu, $$
as for (2.9) as desired. □
Using Lemma 2.3, we immediately obtain the following result.
Lemma 2.4
Suppose
\(K, K_{i}, L, L_{j}\in\mathcal{S}^{n}_{o}\)
and
\(\phi,\phi _{k}\in\Phi\). If
\(K_{i}\rightarrow K\), \(L_{j}\rightarrow L\)
and
\(\phi _{k}\rightarrow\phi\), then
\(\lim_{i,j,k\rightarrow\infty}\widetilde{V}_{-\phi _{k}}(K_{i},L_{j})=\widetilde{V}_{-\phi}(K,L)\).