Harmonic Blaschke–Minkowski Homomorphism

Abstract

Centroid bodies are a continuous and $GL\left(n\right)$-contravariant valuation and play critical roles in the solution to the Busemann–Petty problem. In this paper, we introduce the notion of harmonic Blaschke–Minkowski homomorphism and show that such a map is represented by a spherical convolution operator. Furthermore, we consider the Shephard-type problem of whether $\mathsf{\Phi }K\subseteq \mathsf{\Phi }L$ implies $V\left(K\right)\le V\left(L\right)$, where $\mathsf{\Phi }$ is a harmonic Blaschke–Minkowski homomorphism. Some important results for centroid bodies are extended to a large class of valuations. Finally, we give two interesting results for even and odd harmonic Blaschke–Minkowski homomorphisms, separately.

1. Introduction and Main Results

The setting for this paper is n-dimensional Euclidean spaces ${\mathbb{R}}^n$. Let ${\mathcal{K}}^n$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in ${\mathbb{R}}^n$. ${\mathcal{K}}_0^n$ denotes the set of convex bodies containing the origin in their interiors. A convex body K is determined by its support function $h_K\left(x\right)$, defined on ${\mathbb{R}}^n$ by $h_K\left(x\right)=\max \{x·y:y\in K\},x\in {\mathbb{R}}^n$. The n-dimensional volume of a body K is written as $V\left(K\right)$. The unit sphere in ${\mathbb{R}}^n$ is denoted as ${\mathcal{S}}^{n-1}$, and B denotes the standard unit ball in ${\mathbb{R}}^n$.

If K is a compact star-shaped (about the origin) set in ${\mathbb{R}}^n$, then its radial function, ${\rho }_K=\rho (K,·):{\mathbb{R}}^n\setminus \left\{0\right\}\to [0,\infty )$, is defined by (see [1])

$$\rho (K,u)=max\{\lambda \ge 0,\lambda u\in K\},u\in {\mathcal{S}}^{n-1}.$$

If ${\rho }_K$ is positive and continuous, then K is called a star body (about the origin) and ${\mathcal{S}}^n$ denotes the set of star bodies in ${\mathbb{R}}^n$. We use ${\mathcal{S}}_0^n$ to denote the subset of ${\mathcal{S}}^n$ containing the origin in their interiors. Two star bodies—K and L—are said to be dilations of one another if ${\rho }_K\left(u\right)/{\rho }_L\left(u\right)$ is independent from $u\in {\mathcal{S}}^{n-1}$.

One of the important algebraic structures on the set of convex bodies is the Minkowski addition: for $K,L\in {\mathcal{K}}^n$ and $\lambda ,\mu \ge 0$ (not both zero), the Minkowski linear combination, $\lambda K+\mu L$, of K and L is defined by (see [1])

$$h_{\lambda K+\mu L}\left(u\right)=\lambda h_K\left(u\right)+\mu h_L\left(u\right).$$

(1)

The theory of real valued valuations is at the center of convex geometry. A function $\mathsf{\Phi }$ defined on the space ${\mathcal{S}}^n$ and taking values in an abelian semigroup is called a valuation if for all $K,L\in {\mathcal{S}}^n$ such that $K\cup L\in {\mathcal{S}}^n$ and,

$$\mathsf{\Phi }(K\cup L)+\mathsf{\Phi }(K\cap L)=\mathsf{\Phi }K+\mathsf{\Phi }L.$$

The theory of valuations and its important applications in integral geometry and geometric probability were developed and described in [2,3]. Using the theory of valuations, Schuster introduced the Blaschke–Minkowski homomorphism and radial Blaschke–Minkowski homomorphism (see [4,5,6,7,8]). Recently, Wang extended those definitions to the case of $L_p$ (see [9]).

The notion of Blaschke–Minkowski homomorphism is defined as follows (see [5]).

Definition 1.

A map $\mathsf{\Psi }:{\mathcal{K}}^n\to {\mathcal{K}}^n$ is called Blaschke–Minkowski homomorphism if it satisfies the following conditions:

(1) 

Ψ is continuous.

(2) 

Ψ is Blaschke–Minkowski additive, i.e., $\mathsf{\Psi }(K\mp L)=\mathsf{\Psi }K+\mathsf{\Psi }L$ for all $K,L\in {\mathcal{K}}^n.$

(3) 

Ψ intertwines rotations, i.e., $\mathsf{\Psi }\left(\varphi K\right)=\varphi \mathsf{\Psi }K$, for all $\varphi \in SO\left(n\right)$ and $K\in {\mathcal{K}}^n$.

Here, $\mathsf{\Psi }K+\mathsf{\Psi }L$ denotes the Minkowski sum of $\mathsf{\Psi }K$ and $\mathsf{\Psi }L$, and $K\mp L$ is the Blaschke sum of the convex bodies K and L.

Inspired by Schuster’s idea of Blaschke–Minkowski homomorphism, we find that the centroid operator has a similar structural property, so we introduce the notion of harmonic Blaschke–Minkowski homomorphism.

For $K,L\in {\mathcal{S}}_0^n$ and $\lambda ,\mu \ge 0$ (not both 0), the harmonic Blaschke linear combination (see [10]) $\lambda \star K\widehat{+}\mu \star L$ is

$$\frac{{\rho }^{n+1}(\lambda \star K\widehat{+}\mu \star L)}{V(\lambda \star K\widehat{+}\mu \star L)}=\lambda \frac{{\rho }^{n+1}\left(K\right)}{V\left(K\right)}+\mu \frac{{\rho }^{n+1}\left(L\right)}{V\left(L\right)}.$$

(2)

A centroid body was defined and investigated by Petty (see [11]). For $K\in {\mathcal{S}}_o^n$, the centroid body $\mathsf{\Gamma }K$ of K is an origin-symmetric convex body in which the support function is defined by

$$h_{\mathsf{\Gamma }K}\left(u\right)=\frac{1}{V\left(K\right)}{\int }_K|u·x|dx,$$

where $u\in {\mathcal{S}}^{n-1}$ and $u·x$ denotes the standard inner product of u and x.

A centroid body is a classical notion in geometry and has attracted increased attention in recent years (see [10,12,13,14,15,16,17]). Centroid bodies turned out to be critical for the solution of the Shephard and Shephard problem (see [1,10,11,13]). The fundamental volume inequality for centroid bodies is the Shephard centroid inequality (see [11]): among all of the bodies of a given volume, the centroid bodies have minimal volume precisely for ellipsoids centered in the origin.

The centroid operator $\mathsf{\Gamma }:{\mathcal{S}}_o^n\to {\mathcal{K}}_o^n$ has the following well-known properties:

(a)

$\mathsf{\Gamma }$ is continuous.

(b)

$\mathsf{\Gamma }$ is harmonic Blaschke–Minkowski additive, i.e., $\mathsf{\Gamma }(\lambda \star K\widehat{+}\mu \star L)=\lambda \mathsf{\Gamma }K+\mu \mathsf{\Gamma }L$.

(c)

$\mathsf{\Gamma }$ intertwines rotations, i.e., $\mathsf{\Gamma }\left(\varphi K\right)=\varphi \mathsf{\Gamma }K$, $\varphi \in SO\left(n\right)$.

Here, $\lambda \star K\widehat{+}\mu \star L$ is the harmonic Blaschke sum of the star bodies K and L, and $\lambda \mathsf{\Gamma }K+\mu \mathsf{\Gamma }L$ denotes the Minkowski sum of the centroid bodies $\mathsf{\Gamma }K$ and $\mathsf{\Gamma }L$.

Definition 2.

A map $\mathsf{\Phi }:{\mathcal{S}}_o^n\to {\mathcal{K}}_o^n$ is called a harmonic Blaschke–Minkowski homomorphism if it satisfies the above $\left(a\right)$, $\left(b\right)$, and $\left(c\right)$.

A map $\mathsf{\Phi }:{\mathcal{S}}_o^n\to {\mathcal{K}}_o^n$ is called even if $\mathsf{\Phi }\left(K\right)=\mathsf{\Phi }(-K)$ for all $K\in {\mathcal{S}}_o^n$, and similarly, an odd map means that $\mathsf{\Phi }\left(K\right)=-\mathsf{\Phi }(-K)$. We will give an interesting Theorem (see Theorem 5) for the even and odd harmonic Blaschke–Minkowski homomorphism. From the above definition, we know that the centroid body operator $\mathsf{\Gamma }$ is an example of an even harmonic Blaschke–Minkowski homomorphism. The operator that maps every star body to the origin is called the trivial harmonic Blaschke–Minkowski homomorphism.

The main purpose of this paper is to show that there is a representation for harmonic Blaschke–Minkowski homomorphism analogous to the ones obtained by Schuster (see [4,5,6,7,8]) for Blaschke–Minkowski homomorphism and radial Blaschke–Minkowski homomorphism.

Theorem 1.

If $\mathsf{\Phi }:{\mathcal{S}}_o^n\to {\mathcal{K}}_o^n$ is a harmonic Blaschke–Minkowski homomorphism, then there is a positive $g\in \mathcal{M}(S^{n-1},\widehat{e})$, such that

$$h_{\mathsf{\Phi }K}\left(u\right)=\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\ast g.$$

(3)

Here, the definition of $\mathcal{M}(S^{n-1},\widehat{e})$ will be found in Section 2.

We also consider the Shephard problem for this harmonic Blaschke–Minkowski homomorphism.

Theorem 2.

For $K,L\in {\mathcal{S}}_0^n$, if $\mathsf{\Phi }K\subseteq \mathsf{\Phi }L$, L is the polar of a Blaschke–Minkowski homomorphism; then,

$$V\left(K\right)\le V\left(L\right),$$

(4)

with the equality if and only if $K=L.$

Theorem 3.

For even for harmonic Blaschke–Minkowski homomorphisms Φ and $L\in {\mathcal{S}}_0^n$, if L is not symmetric about the origin, then there exists a symmetric star body K, such that $\mathsf{\Phi }K\subset \mathsf{\Phi }L$, but

$$V\left(K\right)>V\left(L\right).$$

(5)

Moreover, we characterize harmonic Blaschke–Minkowski homomorphism and investigate the Brunn–Minkowski-type inequalities for the harmonic Blaschke combination under this homomorphism.

Theorem 4.

For $K,L\in {\mathcal{S}}_0^n$, $\lambda ,\mu \ge 0$, then

$$V{\left(\mathsf{\Phi }(\lambda \star K\widehat{+}\mu \star L)\right)}^{\frac{1}{n}}\ge \lambda V{\left(\mathsf{\Phi }K\right)}^{\frac{1}{n}}+\mu V{\left(\mathsf{\Phi }L\right)}^{\frac{1}{n}},$$

(6)

with the equality if and only if $\mathsf{\Phi }K$ and $\mathsf{\Phi }L$ are homothetic, and

$$V{\left({\mathsf{\Phi }}^{*}(\lambda \star K\widehat{+}\mu \star L)\right)}^{-\frac{1}{n}}\ge \lambda V{\left({\mathsf{\Phi }}^{*}K\right)}^{-\frac{1}{n}}+\mu V{\left({\mathsf{\Phi }}^{*}L\right)}^{-\frac{1}{n}},$$

(7)

with equality if and only if ${\mathsf{\Phi }}^{*}K$ and ${\mathsf{\Phi }}^{*}L$ are dilates.

Finally, for the even and odd harmonic Blaschke–Minkowski homomorphisms, we spread out the following interesting result.

Theorem 5.

For $K\in {\mathcal{S}}_0^n$ and $\tau \in [-1,1]$, if Φ is an even harmonic Blaschke–Minkowski homomorphism, then

$$V\left(\mathsf{\Phi }\left({\widehat{▿}}^{\tau }K\right)\right)=V\left(\mathsf{\Phi }K\right);$$

(8)

if Φ is an odd harmonic Blaschke–Minkowski homomorphism, then we have

$$V\left(\mathsf{\Phi }K\right)\le V\left({▵}^{\tau }\mathsf{\Phi }K\right)\le V(▵\mathsf{\Phi }K).$$

(9)

If K is not central, there is equality in the left inequality if and only if $\tau =\pm 1$ and equality in the right inequality if and only if $\tau =0$. Here, the definitions of ${▵}^{\tau }K$ and ${\widehat{▿}}^{\tau }K$ are found in Section 2.

2. Preliminaries

If $K_1,K_2,\cdots ,K_n\in {\mathcal{K}}^n$, the mixed volume $V(K_1,K_2,\cdots ,K_n)$ of $K_1,K_2,\cdots ,K_n$ is defined by (see [1])

$$V(K_1,K_2,\cdots ,K_n)=\frac{1}{n}{\int }_{{\mathcal{S}}^{n-1}}{h}_{K_n}\left(u\right)dS(K_1,K_2,\cdots ,K_{n-1},u),$$

(10)

where $S(K_1,K_2,\cdots ,K_{n-1},u)$ is the mixed area measure of $K_1,K_2,\cdots ,K_{n-1}$.

In Equation (10), let ${\underbrace{K_1=\cdots =K_{n-i}}}_{n-i}=K$ and ${\underbrace{K_{n-i+1}=\cdots =K_n}}_i=L(i=0,1,\cdots ,n)$; we denote

$$V(K_1,K_2,\cdots ,K_n)=V(K,n-i;L,i)=V_i(K,L),$$

which is immediately followed by

$$V_1(K,L)=\frac{1}{n}{\int }_{{\mathcal{S}}^{n-1}}{h}_L\left(u\right)dS(K,u).$$

(11)

As the dual case of convex body, a star body $K\in {\mathcal{S}}^n$ is determined by its radial function $\rho (K,u)$ on $S^{n-1}$. From the definition of $\rho (K,u)$, we know that for $\lambda >0$ and $\varphi \in SO\left(n\right)$,

$$\rho (\lambda K,u)=\lambda \rho (K,u)\mathrm{and}\rho (\varphi K,u)=\rho (K,{\varphi }^{-1}u),$$

where ${\varphi }^{-1}$ is the inverse of $\varphi$.

For $K,L\in {\mathcal{S}}^n$ and $\lambda ,\mu \ge 0$, the radial Minkowski linear combination $\lambda K\widetilde{+}\mu L$ is a star body defined by

$$\rho (\lambda K\widetilde{+}\mu L,u)=\lambda \rho (K,u)+\mu \rho (L,u).$$

For $K_1,K_2,\cdots ,K_n\in {\mathcal{S}}^n$, the dual mixed volume $\tilde{V}(K_1,K_2,\cdots ,K_n)$ of $K_1,K_2,\cdots ,K_n$ is defined by (see [18])

$$\tilde{V}(K_1,K_2,\cdots ,K_n)=\frac{1}{n}{\int }_{{\mathcal{S}}^{n-1}}\rho (K_1,u)\rho (K_2,u)\cdots \rho (K_n,u)du.$$

Similarly, we denote

$$\tilde{V}(K_1,K_2,\cdots ,K_n)=\tilde{V}(K,n-i;L,i)=\widetilde{V_i}(K,L).$$

Lutwak in [18] also extended the definition of $\widetilde{V_i}(K,L)$ so that it was defined for all real i as follows:

$$\widetilde{V_i}(K,L)=\frac{1}{n}{\int }_{{\mathcal{S}}^{n-1}}\rho {(K,u)}^{n-i}\rho {(L,u)}^{i}du.$$

(12)

The dual mixed volume has the following well-known harmonic dual Minkowski’s inequality (see [18]). For $K,L\in {\mathcal{S}}_0^n$,

$$V{\left(K\right)}^{\frac{n+1}{n}}V{\left(L\right)}^{\frac{-1}{n}}\le {\tilde{V}}_{-1}(K,L).$$

(13)

Based on the definition of Minkowski linear combination, the difference body, $\mathsf{\Delta }K$ of K, is defined by

$$\mathsf{\Delta }K=\frac{1}{2}K+\frac{1}{2}(-K).$$

The difference body $\mathsf{\Delta }K$ is origin-symmetric. Recently, Wang (see [19]) extended this notion to asymmetric difference body as follows (when $p=1$):

$$\mathsf{\Delta }K=\frac{1+\tau }{2}K+\frac{1+\tau }{2}(-K),$$

(14)

here and throughout $\tau \in [-1,1]$.

For the asymmetric difference body, Wang and Ma in [19] gave the following inequality, which immediately deduces Theorem 5.

Lemma 1.

If $K\in {\mathcal{K}}_0^n$ and $\tau \in [-1,1]$, then

$$V(▵K)\ge V\left({▵}^{\tau }K\right)\ge V\left(K\right).$$

(15)

If K is not central, there is equality in the left inequality if and only if $\tau =0$ and equality in the right inequality if and only if $\tau =\pm 1$.

Lutwak in [10] defined the harmonic Blaschke body as follows:

$$\widehat{▿}K=\frac{1}{2}\star K\widehat{+}\frac{1}{2}\star (-K).$$

The harmonic Blaschke body is origin-symmetric and Lutwak also gave an inequality for $K\in {\mathcal{S}}_0^n$:

$$V\left(\widehat{▿}K\right)\ge V\left(K\right),$$

(16)

with the equality if and only if K is symmetric about the origin.

In 2014, Feng and Wang in [20] extended the notion of harmonic Blaschke body to general harmonic Blaschke body (here, we let $p=1$):

$${\widehat{▿}}^{\tau }K=\frac{1+\tau }{2}\star K\widehat{+}\frac{1-\tau }{2}\star (-K).$$

(17)

The asymmetric difference body and the general harmonic Blaschke body belong to a new and rapidly evolving asymmetric $L_p$-Brunn–Minkowski theory that has its origins in the work of Ludwig, Haberl, and Schuster (see [21,22,23,24,25]). For further research on asymmetric $L_p$-Brunn-Minkowski theory, see also [19,20,26,27,28,29,30,31,32,33,34,35].

Some basic notions on spherical harmonics will be required (see [4]). Let $\mathcal{C}\left(SO\right(n\left)\right)$ denote the set of continuous functions on $SO\left(n\right)$ with the uniform topology and $\mathcal{M}\left(SO\right(n\left)\right)$ denote its dual space of signed finite measures on $SO\left(n\right)$ with the weak topology. ${\mathcal{M}}_{+}\left(SO\left(n\right)\right)$ is the set of nonnegative measures on $SO\left(n\right)$. For $\mu \in \mathcal{M}\left(SO\right(n\left)\right)$ and $f\in \mathcal{C}\left(SO\right(n\left)\right)$, the canonical pairing is

$$⟨\mu ,f⟩=⟨f,\mu ⟩={\int }_{SO\left(n\right)}f\left(\varphi \right)d\mu \left(\varphi \right).$$

Sometimes, we identify a continuous function f with the absolute continuous measure with density f and thus view $\mathcal{C}\left(SO\right(n\left)\right)$ as a subspace of $\mathcal{M}\left(SO\right(n\left)\right)$. The canonical pairing is then consistent with the usual inner product on $\mathcal{C}\left(SO\right(n\left)\right)$.

$SO\left(n\right)$ denotes the subgroup of rotations leaving the pole $\widehat{e}$ of ${\mathcal{S}}^{n-1}$ fixed, and the sphere ${\mathcal{S}}^{n-1}$ is identified with the homogeneous space $SO\left(n\right)/SO(n-1)$:

$$u=\varphi \widehat{e}\mapsto \varphi SO(n-1).$$

The projection from $SO\left(n\right)$ onto ${\mathcal{S}}^{n-1}$ is $\varphi \mapsto \widehat{\varphi }:=\varphi \widehat{e}.$ The unity $e\in SO\left(n\right)$ is mapped to the pole of the sphere $\widehat{e}\in {\mathcal{S}}^{n-1}$. We call a measure $\mu \in \mathcal{M}\left(SO\right(n\left)\right)$ zonal if $\varphi \mu =\mu$ for every $\varphi \in SO(n-1)$. The set of all continuous, zonal functions is denoted by $\mathcal{C}({\mathcal{S}}^{n-1},\widehat{e})$ and $\mathcal{M}({\mathcal{S}}^{n-1},\widehat{e})$ denotes the set of zonal measures on ${\mathcal{S}}^{n-1}$.

The convolution $\mu \ast f\in \mathcal{C}\left({\mathcal{S}}^{n-1}\right)$ of a measure $\mu \in \mathcal{M}\left(SO\right(n\left)\right)$ and a function $f\in \mathcal{C}\left({\mathcal{S}}^{n-1}\right)$ is defined by (see [4])

$$(\mu \ast f)\left(u\right)={\int }_{SO\left(n\right)}\varphi f\left(u\right)d\mu \left(\varphi \right).$$

We lay out the following Lemma 2 (see [5]), which is needed in the proof of Theorem 1.

Lemma 2.

A map $\mathsf{\Psi }:C(S^{n-1}\to C\left(S^{n-1}\right)$ is a monotone, linear map that intertwines rotations if and only if there is a measure $\mu \in {\mathcal{M}}_{+}(S^{n-1},\widehat{e})$ such that

$$\mathsf{\Psi }f=f\ast \mu .$$

3. Proofs of Main Results

In this section, we prove the results listed in Section 1.

From Definition Section 1, we can obtain that the well-known centroid operator is a harmonic Blaschke–Minkowski homomorphism. Let $g=\frac{|u·v|}{n+1}$; then, $g\in \mathcal{C}(S^{n-1},\widehat{e})\subseteq \mathcal{M}(S^{n-1},\widehat{e})$, and we obtain that the definition of centroid body $\mathsf{\Gamma }$ is

$$h_{\mathsf{\Gamma }K}\left(u\right)=\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\ast g=\frac{1}{(n+1)V\left(K\right)}{\int }_{S^{n-1}}|u·v|{\rho }_K^{n+1}\left(v\right)dv.$$

Proof of Theorem 1. 

Suppose that a map $\mathsf{\Phi }:{\mathcal{S}}_o^n\to {\mathcal{K}}_o^n$ satisfies $h_{\mathsf{\Phi }K}\left(u\right)=\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\ast g$, where $g\in \mathcal{M}(S^{n-1},\widehat{e})$ is a nonnegative measure. We prove such a map $\mathsf{\Phi }$ is a harmonic Blaschke–Minkowski homomorphism. The continuity of $\mathsf{\Phi }$ follows from the fact that the support function $h_K\left(u\right)$ is continuous with respect to Hausdorff metric. Furthermore, the following means that $\mathsf{\Phi }$ satisfies the harmonic Blaschke–Minkowski additive:

$$\begin{array}{cc}\hfill h_{\mathsf{\Phi }K+\mathsf{\Phi }L}\left(u\right)& =h_{\mathsf{\Phi }K}\left(u\right)+h_{\mathsf{\Phi }L}\left(u\right)\hfill \\ & =\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\ast g+\frac{{\rho }_L^{n+1}\left(v\right)}{V\left(L\right)}\ast g\hfill \\ & =(\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}+\frac{{\rho }_L^{n+1}\left(v\right)}{V\left(L\right)})\ast g\hfill \\ & =\frac{{\rho }_{K\widehat{+}L}^{n+1}\left(v\right)}{V\left(K\widehat{+}L\right)}\ast g\hfill \\ & =h_{\mathsf{\Phi }\left(K\widehat{+}L\right)}\left(u\right).\hfill \end{array}$$

that is

$$\mathsf{\Phi }\left(K\widehat{+}L\right)=\mathsf{\Phi }K+\mathsf{\Phi }L.$$

Additionally, for every $\varphi \in SO\left(n\right)$,

$$h_{\mathsf{\Phi }\left(\varphi k\right)}\left(u\right)=\frac{{\rho }_{\varphi K}^{n+1}\left(v\right)}{V\left(\varphi K\right)}\ast g=\frac{{\rho }_K^{n+1}\left({\varphi }^{-1}v\right)}{V\left(K\right)}\ast g=h_{\mathsf{\Phi }\left(k\right)}\left({\varphi }^{-1}u\right)=h_{\varphi \mathsf{\Phi }\left(k\right)}\left(u\right)$$

which means $\mathsf{\Phi }\left(\varphi K\right)=\varphi \mathsf{\Phi }K$.

Thus, the map $\mathsf{\Phi }$ that satisfies $h_{\mathsf{\Phi }K}\left(u\right)=\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\ast g$ is a harmonic Blaschke–Minkowski homomorphism (satisfies the properties $\left(a\right)$, $\left(b\right)$ and $\left(c\right)$ of Definition 1). Next, we have to show that for every such operator $\mathsf{\Phi }$, there is a nonnegative measure $g\in \mathcal{M}(S^{n-1},\widehat{e})$ such that Equation (1) holds.

Since every positive continuous function on ${\mathcal{S}}^{n-1}$ is a radial function of some star body, the vector space $\{\frac{{\rho }_{K_1}^{n+1}\left(v\right)}{V\left(K_1\right)}-\frac{{\rho }_{K_2}^{n+1}\left(v\right)}{V\left(K_2\right)}:K,L\in {\mathcal{S}}_o^n\}$ coincides with $\mathcal{M}(S^{n-1},\widehat{e})$. Define $\overline{\mathsf{\Phi }}:C\left(S^{n-1}\right)\to C\left(S^{n-1}\right)$ as follows:

$$\overline{\mathsf{\Phi }}{\mu }_1=h_{\mathsf{\Phi }{K}_1}\left(u\right)-h_{\mathsf{\Phi }{K}_2}\left(u\right),$$

where

$${\mu }_1=\frac{{\rho }_{K_1}^{n+1}\left(v\right)}{V\left(K_1\right)}-\frac{{\rho }_{K_2}^{n+1}\left(v\right)}{V\left(K_2\right)}.$$

(18)

The operator $\overline{\mathsf{\Phi }}$ for ${\mu }_2=\frac{{\rho }_{L_1}^{n+1}\left(v\right)}{V\left(L_1\right)}-\frac{{\rho }_{L_2}^{n+1}\left(v\right)}{V\left(L_2\right)}$ immediately yields:

$$\overline{\mathsf{\Phi }}{\mu }_2=h_{\mathsf{\Phi }{L}_1}\left(u\right)-h_{\mathsf{\Phi }{L}_2}\left(u\right).$$

(19)

Considering that Equations (16) and (19), and $\mathsf{\Phi }$ are harmonic Blaschke–Minkowski homomorphisms, we have

$$\begin{array}{cc}\hfill \overline{\mathsf{\Phi }}{\mu }_1+\overline{\mathsf{\Phi }}{\mu }_2& =h_{\mathsf{\Phi }{K}_1}\left(u\right)-h_{\mathsf{\Phi }{K}_2}\left(u\right)+h_{\mathsf{\Phi }{L}_1}\left(u\right)-h_{\mathsf{\Phi }{L}_2}\left(u\right)\hfill \\ & =h_{\mathsf{\Phi }{K}_1+\mathsf{\Phi }{L}_1}\left(u\right)-h_{\mathsf{\Phi }{K}_2+\mathsf{\Phi }{L}_2}\left(u\right)\hfill \\ & =h_{\mathsf{\Phi }\left(K_1\widehat{+}{L}_1\right)}\left(u\right)-h_{\mathsf{\Phi }\left(K_2\widehat{+}{L}_2\right)}\left(u\right)\hfill \\ & =\overline{\mathsf{\Phi }}(\frac{{\rho }_{K_1\widehat{+}{L}_1}^{n+1}\left(v\right)}{V\left(K_1\widehat{+}{L}_1\right)}-\frac{{\rho }_{K_2\widehat{+}{L}_2}^{n+1}\left(v\right)}{V\left(K_2\widehat{+}{L}_2\right)})\hfill \\ & =\overline{\mathsf{\Phi }}({\mu }_1+{\mu }_2).\hfill \end{array}$$

which means that the map $\overline{\mathsf{\Phi }}$ is linear. Therefore, $\overline{\mathsf{\Phi }}$ is a linear extension of $\mathsf{\Phi }$ to $\mathcal{C}\left(S^{n-1}\right)$ and $\overline{\mathsf{\Phi }}$ intertwines rotations. Since the cone of radial functions is invariant under $\overline{\mathsf{\Phi }}$, it is also monotone. Hence, by Lemma 1, there is a measure $g\in \mathcal{M}(S^{n-1},\widehat{e})$ such that $\overline{\mathsf{\Psi }}f=f\ast g$, which leads to

$$\overline{\mathsf{\Phi }}\left(\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\right)=\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(K\right)}\ast g=h_{\mathsf{\Phi }K}\left(u\right).$$

□

Before the proof of Theorem 2, we reveal the relationship between harmonic Blaschke–Minkowski homomorphism and Blaschke–Minkowski homomorphism, which are needed in the proof of Theorem 2 as Lemma 3 followed immediately.

Lemma 3.

For any harmonic Blaschke–Minkowski homomorphism Φ and $M\in {\mathcal{K}}_o^n,N\in {\mathcal{S}}_0^n$, we have

$$V_1(M,\mathsf{\Phi }N)=\frac{1}{V\left(N\right)}{\tilde{V}}_{-1}(N,{\mathsf{\Psi }}^{*}M).$$

Here, ${\mathsf{\Psi }}^{*}$ is the polar of Blaschke–Minkowski homomorphism Ψ.

Proof. 

Let a nonnegative $g\in \mathcal{M}(S^{n-1},\widehat{e})$ be the generating measure of $\mathsf{\Phi }$. Considering Equation (13) and Theorem 1, we have

$$\begin{array}{cc}\hfill V_1(M,\mathsf{\Phi }N)& =\frac{1}{n}{\int }_{S^{n-1}}{h}_{\mathsf{\Phi }N}\left(u\right)dS(M,u)\hfill \\ & =\frac{1}{n}{\int }_{S^{n-1}}[{\int }_{S^{n-1}}\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(N\right)}·g(u,v)dv]dS(M,u)\hfill \\ & =\frac{1}{n}{\int }_{S^{n-1}}[{\int }_{S^{n-1}}g(u,v)dS(M,u)]\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(N\right)}dv\hfill \\ & =\frac{1}{n}{\int }_{S^{n-1}}{h}_{\mathsf{\Psi }M}\left(u\right)\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(N\right)}dv\hfill \\ & =\frac{1}{n}{\int }_{S^{n-1}}\frac{{\rho }_K^{n+1}\left(v\right)}{V\left(N\right)}{\rho }_{{\mathsf{\Psi }}^{*}M\left(v\right)}^{-1}dv\hfill \\ & =\frac{1}{V\left(N\right)}{\tilde{V}}_{-1}(N,{\mathsf{\Psi }}^{*}M).\hfill \end{array}$$

□

Proof of Theorem 2. 

Since $\mathsf{\Phi }K\subset \mathsf{\Phi }L$, for all $M\in {\mathcal{K}}^n$, we have

$$V_1(M,\mathsf{\Phi }K)\le V_1(M,\mathsf{\Phi }L).$$

Considering Lemma 3, there is a Blaschke–Minkowski homomorphism $\mathsf{\Psi }$, such that

$$\frac{{\tilde{V}}_{-1}(K,{\mathsf{\Psi }}^{*}M)}{V\left(K\right)}\le \frac{{\tilde{V}}_{-1}(L,{\mathsf{\Psi }}^{*}M)}{V\left(L\right)}$$

On the other hand, L is the polar of a Blaschke–Minkowski homomorphism, so if we let $L={\mathsf{\Psi }}^{*}M$, then

$$\frac{{\tilde{V}}_{-1}(K,L)}{V\left(K\right)}\le \frac{{\tilde{V}}_{-1}(L,L)}{V\left(L\right)}=1,$$

which means

$${\tilde{V}}_{-1}(K,L)\le V\left(K\right).$$

Thus, Equation (13) leads to

$$V{\left(K\right)}^{\frac{n+1}{n}}V{\left(L\right)}^{\frac{-1}{n}}\le V\left(K\right)$$

i.e.,

$$V\left(K\right)\le V\left(L\right).$$

From the equality condition of Equation (13), we know that equality holds if and only if K and L are dilates of each other. □

Proof of Theorem 3. 

Since L is not symmetric about the origin, we have

$$V\left(\widehat{▿}L\right)>V\left(L\right).$$

Let $0<r<1$, such that $V\left(r\widehat{▿}L\right)>V\left(L\right)$, and let $K=r\widehat{▿}L$; then, K is origin-symmetric and

$$\mathsf{\Phi }K=\mathsf{\Phi }\left(r\widehat{▿}L\right)=r\mathsf{\Phi }\left(\widehat{▿}L\right)=r\frac{1}{2}\mathsf{\Phi }L+r\frac{1}{2}\mathsf{\Phi }(-L).$$

Considering that $\mathsf{\Phi }$ is an even harmonic Blaschke–Minkowski homomorphism, we have

$$\mathsf{\Phi }K=r\mathsf{\Phi }L\subset \mathsf{\Phi }L,$$

but

$$V\left(K\right)>V\left(L\right).$$

□

Proof of Theorem 4. 

Considering the definitions of harmonic Blaschke–Minkowski homomorphism and Brunn–Minkowski inequality, Equation (6) holds immediately.

For Equation (7), we can see that it is just the special case $i=0$ of the general inequality for dual quermassintegrals ${\tilde{W}}_i\left({\mathsf{\Phi }}^{*}(\lambda \star K\widehat{+}\mu \star L)\right),$ for $0\le i\le n-1$:

$$\begin{array}{cc}& {\tilde{W}}_i{\left({\mathsf{\Phi }}^{*}(\lambda \star K\widehat{+}\mu \star L)\right)}^{-\frac{1}{n-i}}\hfill \\ & =[\frac{1}{n}{\int }_{S^{n-1}}{\rho }^{n-i}{({\mathsf{\Phi }}^{*}(\lambda \star K\widehat{+}\mu \star L,u)du]}^{-\frac{1}{n-i}}\hfill \\ & =[\frac{1}{n}{\int }_{S^{n-1}}{h}^{-n+i}{(\mathsf{\Phi }(\lambda \star K\widehat{+}\mu \star L,u)du]}^{-\frac{1}{n-i}}\hfill \\ & ={[\frac{1}{n}{\int }_{S^{n-1}}{\left(\lambda h(\mathsf{\Phi }K,u)+\mu h(\mathsf{\Phi }L,u)\right)}^{-n+i}du]}^{-\frac{1}{n-i}}\hfill \\ & \ge {[\frac{1}{n}{\int }_{S^{n-1}}{\left(\lambda h(\mathsf{\Phi }K,u)\right)}^{-n+i}du]}^{-\frac{1}{n-i}}+{[\frac{1}{n}{\int }_{S^{n-1}}{\left(\mu h(\mathsf{\Phi }L,u)\right)}^{-n+i}du]}^{-\frac{1}{n-i}}\hfill \\ & ={[\frac{1}{n}{\int }_{S^{n-1}}{\left(\lambda \rho ({\mathsf{\Phi }}^{*}K,u)\right)}^{n-i}du]}^{-\frac{1}{n-i}}+{[\frac{1}{n}{\int }_{S^{n-1}}{\left(\mu \rho ({\mathsf{\Phi }}^{*}L,u)\right)}^{n-i}du]}^{-\frac{1}{n-i}}\hfill \\ & ={\tilde{W}}_i{\left({\mathsf{\Phi }}^{*}K\right)}^{-\frac{1}{n-i}}+{\tilde{W}}_i{\left({\mathsf{\Phi }}^{*}L\right)}^{-\frac{1}{n-i}}.\hfill \end{array}$$

From the equality condition of Minkowskis inequality, we see that equality holds in above inequality if and only if ${\mathsf{\Phi }}^{*}K$ and ${\mathsf{\Phi }}^{*}L$ are dilates. □

Finally, we give the proof for Theorem 5.

Proof of Theorem 5. 

From (17) and the definition of harmonic Blaschke–Minkowski homomorphism, we immediately obtain the following result:

$$\mathsf{\Phi }\left({\widehat{▿}}^{\tau }K\right)=\frac{1+\tau }{2}\mathsf{\Phi }K+\frac{1-\tau }{2}\mathsf{\Phi }(-K),$$

which tell us that if $\mathsf{\Phi }$ is even. Then,

$$\mathsf{\Phi }\left({\widehat{▿}}^{\tau }K\right)=\mathsf{\Phi }K,$$

and if $\mathsf{\Phi }$ is odd, we have

$$\mathsf{\Phi }\left({\widehat{▿}}^{\tau }K\right)=\frac{1+\tau }{2}\mathsf{\Phi }K+\frac{1-\tau }{2}(-\mathsf{\Phi }K),$$

(20)

The right-hand side of Equation (20) is the asymmetric difference body of $\mathsf{\Phi }K$ when $p=1$ (see [19]), thus by the above Lemma 1, Theorem 5 is proven immediately. □

4. Conclusions

In this manuscript, we introduced and discussed the notion of harmonic Blaschke–Minkowski homomorphism based on the Centroid bodies. Under this harmonic Blaschke–Minkowski homomorphism, several results of the Centroid bodies can be extended to a large class of valuations. Some questions still remain to be studied deeply. In the future, we will research further the centroid bodies in this harmonic Blaschke–Minkowski homomorphism.