$L_p$ Blaschke-Minkowski homomorphisms

Abstract

In this paper, we introduce the concept of $L_p$ Blaschke-Minkowski homomorphisms and show that those maps are represented by a spherical convolution operator. And then we consider the Busemann-Petty type problem for $L_p$ Blaschke-Minkowski homomorphisms.

MSC:52A40, 52A20.

1 Introduction

The theory of real valued valuations is at the center of convex geometry. Blaschke started a systematic investigation in the 1930s, and then Hadwiger [1] focused on classifying valuations on compact convex sets in ${\mathbb{R}}^n$ and obtained the famous Hadwiger’s characterization theorem. Schneider [2] obtained first results on convex body valued valuations with Minkowski addition in 1970s. The survey [3] and the book [4] are an excellent source for the classical theory of valuations. Some more recent results can see [1, 5–20].

An operator $Z:{\mathcal{K}}^n\to {\mathcal{K}}^n$ is called a Minkowski valuation if

$$Z(K\cup L)+Z(K\cap L)=ZK+ZL,$$

(1.1)

whenever $K,L,K\cup L\in {\mathcal{K}}^n$, and here + is the Minkowski addition.

A Minkowski valuation Z is called $SO(n)$ equivariant, if for all $\vartheta \in SO(n)$ and all $K\in {\mathcal{K}}^n$,

$$Z(\vartheta K)=\vartheta ZK.$$

(1.2)

A Minkowski valuation Z is called homogeneity of degree p, if for all $K\in {\mathcal{K}}^n$ and all $\lambda \ge 0$,

$$Z(\lambda K)={\lambda }^{p}ZK.$$

(1.3)

A map $\mathrm{\Phi }:{\mathcal{K}}^n\to {\mathcal{K}}^n$ is called a Blaschke-Minkowski homomorphism if it is continuous, $SO(n)$ equivariant and satisfies $\mathrm{\Phi }(K\mathrm{\#}L)=\mathrm{\Phi }K+\mathrm{\Phi }L$, where # denotes the Blaschke addition, i.e., $S(K\mathrm{\#}L,\cdot )=S(K,\cdot )+S(L,\cdot )$.

Obviously, a Blaschke-Minkowski homomorphism is a continuous Minkowski valuation which is $SO(n)$ equivariant and $(n-1)$-homogeneous. Schuster introduced Blaschke-Minkowski homomorphisms and studied the Busemann-Petty type problem for them.

Theorem A [15]

If $\mathrm{\Phi }:{\mathcal{K}}^n\to {\mathcal{K}}^n$ be a Blaschke-Minkowski homomorphism, then there is a weakly positive $g\in \mathcal{C}(S^{n-1},\hat{e})$, unique up to a linear function, such that

$$h(\mathrm{\Phi }K,\cdot )=S(K,\cdot )\ast g.$$

Theorem B [16]

Let $\mathrm{\Phi }:{\mathcal{K}}^n\to {\mathcal{K}}^n$ be a Blaschke-Minkowski homomorphism. If $K\in \mathrm{\Phi }{\mathcal{K}}^n$ and $L\in {\mathcal{K}}^n$, then

$$\mathrm{\Phi }K\subseteq \mathrm{\Phi }L\Rightarrow V(K)\le V(L),$$

and $V(K)=V(L)$ if and only if $K=L$.

Recently, the investigations of convex body and star body valued valuations have received great attention from a series of articles by Ludwig [10–13]; see also [8]. She started systematic studies and established complete classifications of convex and star body valued valuations with respect to $L_p$ Minkowski addition and $L_p$ radial which are compatible with the action of the group $GL(n)$. Based on these results, in this article we study $L_p$ Blaschke-Minkowski homomorphisms which are continuous, $(\frac{n}{p}-1)$-homogeneous and $SO(n)$ equivariant.

Theorem 1.1 Let $p>1$ and $p\ne n$. If ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ be an $L_p$ Blaschke-Minkowski homomorphism, then there is a nonnegative function $g\in \mathcal{C}(S^{n-1},\hat{e})$, such that

$$h^p({\mathrm{\Phi }}_{p}K,\cdot )=S_p(K,\cdot )\ast g.$$

(1.4)

Theorem 1.2 Let $1<p<n$ and p is not an even integer, and let ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ be an $L_p$ Blaschke-Minkowski homomorphism. If $K\in {\mathcal{K}}_e^n$ and $L\in {\mathrm{\Phi }}_p{\mathcal{K}}_e^n$, then

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\Rightarrow V(K)\le V(L).$$

(1.5)

If $p>n$ and p is not an even integer, then

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\Rightarrow V(K)\ge V(L),$$

(1.6)

and $V(K)=V(L)$, if and only if $K=L$.

2 Notation and background material

Let ${\mathcal{K}}_0^n$ denote the set of convex bodies containing the origin in their interiors, and let ${\mathcal{K}}_e^n$ denote origin-symmetric convex bodies. In this paper, we restrict the dimension of ${\mathbb{R}}^n$ to $n\ge 3$. A convex body $K\in {\mathcal{K}}^n$ is uniquely determined by its support function, $h(K,\cdot )$. From the definition of $h(K,\cdot )$, it follows immediately that for $\lambda >0$ and $\vartheta \in SO(n)$,

$$h(\lambda K,u)=\lambda h(K,u)\text{and}h(\vartheta K,u)=h(K,{\vartheta }^{-1}u),$$

(2.1)

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

For $K,L\in {\mathcal{K}}_0^n$, $p\ge 1$, and $\epsilon >0$, the $L_p$ Minkowski addition $K{+}_p\epsilon \cdot L\in {\mathcal{K}}_0^n$ is defined by (see [21])

$$h{(K{+}_p\epsilon \cdot L,\cdot )}^p=h{(K,\cdot )}^p+\epsilon h{(L,\cdot )}^p,$$

(2.2)

where ‘ ⋅ ’ in $\epsilon \cdot L$ denotes the Firey scalar multiplication, i.e., $\epsilon \cdot L={\epsilon }^{\frac{1}{p}}L$.

If $K,L\in {\mathcal{K}}_0^n$, then for $p\ge 1$, the $L_p$ mixed volume, $V_p(K,L)$, of K and L is defined by (see [21])

$$V_p(K,L)=\underset{\epsilon \to 0^{+}}{lim}\frac{V(K{+}_p\epsilon \cdot L)-V(K)}{\epsilon }.$$

Corresponding to each $K\in {\mathcal{K}}_0^n$, there is a positive Borel measure, $S_p(K,\cdot )$, on $S^{n-1}$ such that (see [21])

$$V_p(K,L)=\frac{1}{n}{\int }_{S^{n-1}}h{(L,u)}^{p}d{S}_p(K,u),$$

(2.3)

for each $L\in {\mathcal{K}}_0^n$. The measure $S_p(K,\cdot )$ is just the $L_p$ surface area measure of K, which is absolutely continuous with respect to classical surface area measure $S(K,\cdot )$, and has a Radon-Nikodym derivative

$$\frac{d{S}_p(K,\cdot )}{dS(K,\cdot )}=h{(K,\cdot )}^{1-p}.$$

(2.4)

A convex body $K\in {\mathcal{K}}_0^n$ is said to have a p-curvature function (see [21]) $f_p(K,\cdot ):S^{n-1}\to \mathbb{R}$, if its $L_p$ surface area measure $S_p(K,\cdot )$ is absolutely continuous with respect to spherical Lebesgue measure S and the Radon-Nikodym derivative

$$\frac{d{S}_p(K,\cdot )}{dS}=f_p(K,\cdot ).$$

(2.5)

From the formula (2.3), it follows immediately that for each $K\in K_0^n$,

$$V_p(K,K)=V(K).$$

The Minkowski inequality for the $L_p$ mixed volume states that (see [21]): For $K,L\in {\mathcal{K}}_0^n$, if $p\ge 1$, then

$$V_p(K,L)\ge V{(K)}^{\frac{n-p}{n}}V{(L)}^{\frac{p}{n}},$$

(2.6)

if $p>1$, equality holds if and only if K and L are dilates; if $p=1$, equality holds if and only if K and L are homothetic.

The $L_p$ Minkowski problem asks for necessary and sufficient conditions for a Borel measure μ on $S^{n-1}$ to be the $L_p$ surface area measure of a convex body. Lutwak [22] gave a weak solution to the $L_p$ Minkowski problem as follows.

Theorem C If μ is an even position Borel measure on $S^{n-1}$, which is not concentrated on any great subsphere, then for any $p>1$ and $p\ne n$, there exists a unique origin-symmetric convex bodies $K\in {\mathcal{K}}_e^n$, such that

$$S_p(K,\cdot )=\mu .$$

From (2.4), for $\lambda >0$, we have

$$S_p(\lambda K,\cdot )={\lambda }^{n-p}{S}_p(K,\cdot ).$$

(2.7)

Noting the fact $S(\vartheta K,\cdot )=\vartheta S(K,\cdot )$ for $\vartheta \in SO(n)$ and (2.1), one can obtain

$$S_p(\vartheta K,\cdot )=\vartheta S_p(K,\cdot ),$$

(2.8)

where $\vartheta S_p(K,\cdot )$ is the image measure of $S_p(K,\cdot )$ under the rotation ϑ. Obviously, $S_1(K,\cdot )$ is just $S(K,\cdot )$.

The $L_p$ Blaschke addition $K{\mathrm{\#}}_{p}L$ of $K,L\in {\mathcal{K}}_0^n$ is the convex body with

$$S_p(K{\mathrm{\#}}_{p}L,\cdot )=S_p(K,\cdot )+S_p(L,\cdot ).$$

(2.9)

Some basic notions on spherical harmonics will be required. The article by Grinberg and Zhang [23] and the article by Schuster [16] are excellent general references on spherical harmonics. As usual, $SO(n)$ and $S^{n-1}$ will be equipped with the invariant probability measures. Let $\mathcal{C}(SO(n))$, $\mathcal{C}(S^{n-1})$ be the spaces of continuous functions on $SO(n)$ and $S^{n-1}$ with uniform topology and $\mathcal{M}(SO(n))$, $\mathcal{M}(S^{n-1})$ their dual spaces of signed finite Borel measures with weak^∗ topology. The group $SO(n)$ acts on these spaces by left translation, i.e., for $f\in \mathcal{C}(S^{n-1})$ and $\mu \in \mathcal{M}(S^{n-1})$, we have $\vartheta f(u)=f({\vartheta }^{-1}u)$, $\vartheta \in SO(n)$, and ϑμ is the image measure of μ under the rotation ϑ.

The sphere $S^{n-1}$ is identified with the homogeneous space $SO(n)/SO(n-1)$, where $SO(n-1)$ denotes the subgroup of rotations leaving the pole $\hat{e}$ of $S^{n-1}$ fixed. The projection from $SO(n)$ onto $S^{n-1}$ is $\vartheta \mapsto \hat{\vartheta }:=\vartheta \hat{e}$. Functions on $S^{n-1}$ can be identified with right $SO(n-1)$-invariant functions on $SO(n)$, by $\check{f}(\vartheta )=f(\hat{\vartheta })$, for $f\in \mathcal{C}(S^{n-1})$. In fact, $\mathcal{C}(S^{n-1})$ is isomorphic to the subspace of right $SO(n-1)$-invariant functions in $\mathcal{C}(SO(n))$.

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

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

(2.10)

The canonical pairing of $f\in \mathcal{C}(S^{n-1})$ and $\mu \in \mathcal{M}(S^{n-1})$ is defined by

$$〈\mu ,f〉=〈f,\mu 〉={\int }_{S^{n-1}}f(u)d\mu (u).$$

(2.11)

A function $f\in \mathcal{C}(S^{n-1})$ is called zonal, if $\vartheta f=f$ for every $\vartheta \in SO(n-1)$. Zonal functions depend only on the value $u\cdot \hat{e}$. The set of continuous zonal functions on $S^{n-1}$ will be denoted by $\mathcal{C}(S^{n-1},\hat{e})$ and the definition of $\mathcal{M}(S^{n-1},\hat{e})$ is analogous. A map $\mathrm{\Lambda }:\mathcal{C}[-1,1]\to \mathcal{C}(S^{n-1},\hat{e})$ is defined by

$$\mathrm{\Lambda }f(u)=f(u\cdot \hat{e}),u\in S^{n-1}.$$

(2.12)

The map Λ is also an isomorphism between functions on $[-1,1]$ and zonal functions on $S^{n-1}$. If $f\in \mathcal{C}(S^{n-1})$, $\mu \in \mathcal{M}(S^{n-1},\hat{e})$ and $\eta \in SO(n)$, then

$$(f\ast \mu )(\hat{\eta })={\int }_{S^{n-1}}f(\eta u)d\mu (u).$$

(2.13)

If $\mu \in \mathcal{M}(S^{n-1},\hat{e})$, for each $f\in \mathcal{C}(S^{n-1})$ and every $\vartheta \in SO(n)$, then

$$(\vartheta f)\ast \mu =\vartheta (f\ast \mu ).$$

(2.14)

We denote ${\mathcal{H}}_k^n$ by the finite dimensional vector space of spherical harmonics of dimension n and order k, and let $N(n,k)$ be the dimension of ${\mathcal{H}}_k^n$. The space of all finite sums of spherical harmonics of dimension n is denoted by ${\mathcal{H}}^n$. The spaces ${\mathcal{H}}_k^n$ are pairwise orthogonal with respect to the usual inner product on $\mathcal{C}(S^{n-1})$. Clearly, ${\mathcal{H}}_k^n$ is invariant with respect to rotations.

Let $P_k^n\in \mathcal{C}[-1,1]$ denote the Legendre polynomial of dimension n and order k. The zonal function $\mathrm{\Lambda }{P}_k^n$ is up to a multiplicative constant the unique zonal spherical harmonic in ${\mathcal{H}}_k^n$. In each space ${\mathcal{H}}_k^n$ we choose an orthonormal basis $H_{k1},\dots ,H_{kN(n,k)}$. The collection $\{H_{k1},\dots ,H_{kN(n,k)}:k\in \mathbb{N}\}$ forms a complete orthogonal system in ${\mathcal{L}}^2(S^{n-1})$. In particular, for every $f\in {\mathcal{L}}^2(S^{n-1})$, the series

$$f\sim \sum _{k=0}^{\mathrm{\infty }}{\pi }_{k}f$$

converges to f in the ${\mathcal{L}}^2(S^{n-1})$-norm, where ${\pi }_{k}f\in {\mathcal{H}}_k^n$ is the orthogonal projection of f on the space ${\mathcal{H}}_k^n$. Using well-known properties of the Legendre polynomials, it is not hard to show that

$${\pi }_{k}f=N(n,k)(f\ast \mathrm{\Lambda }{P}_k^n).$$

(2.15)

This leads to the spherical expansion of a measure $\mu \in \mathcal{M}(S^{n-1})$,

$$\mu \sim \sum _{k=0}^{\mathrm{\infty }}{\pi }_k\mu ,$$

(2.16)

where ${\pi }_k\mu \in {\mathcal{H}}_k^n$ is defined by

$${\pi }_k\mu =N(n,k)(\mu \ast \mathrm{\Lambda }{P}_k^n).$$

(2.17)

From $P_0^n(t)=1$, $N(n,0)=1$ and $P_1^n(t)=t$, $N(n,1)=n$, we obtain, for $\mu \in \mathcal{M}(S^{n-1})$, the following special cases of (2.18):

$${\pi }_0\mu =\mu \left(S^{n-1}\right)\text{and}({\pi }_1\mu )(u)=n{\int }_{S^{n-1}}u\cdot vd\mu (v).$$

(2.18)

Let ${\kappa }_n$ denote the volume of the Euclidean unit ball B. By (2.3) and (2.19), for every convex body $K\in {\mathcal{K}}_0^n$, it follows that

$${\kappa }_n{\pi }_{0}h{(K,\cdot )}^p=V_p(B,K)\text{and}{\pi }_0{S}_p(K,\cdot )=n{V}_p(K,B).$$

(2.19)

A measure $\mu \in \mathcal{M}(S^{n-1})$ is uniquely determined by its series expansion (2.19). Using the fact that $\mathrm{\Lambda }{P}_k^n$ is (essentially) the unique zonal function in ${\mathcal{H}}_k^n$, a simple calculation shows that for $\mu \in \mathcal{M}(S^{n-1},\hat{e})$, formula (2.18) becomes

$${\pi }_k\mu =N(n,k)〈\mu ,\mathrm{\Lambda }{P}_k^n〉\mathrm{\Lambda }{P}_k^n.$$

(2.20)

A zonal measure $\mu \in \mathcal{M}(S^{n-1},\hat{e})$ is defined by its so-called Legendre coefficients ${\mu }_k:=〈\mu ,\mathrm{\Lambda }{P}_k^n〉$. Using ${\pi }_{k}H=H$ for every $H\in {\mathcal{H}}_k^n$ and the fact that spherical convolution of zonal measures is commutative, we have the Funk-Hecke theorem: If $\mu \in \mathcal{M}(S^{n-1},\hat{e})$ and $H\in {\mathcal{H}}_k^n$, then $H\ast \mu ={\mu }_{k}H$.

A map $\mathrm{\Phi }:\mathcal{D}\subseteq \mathcal{M}(S^{n-1})\to \mathcal{M}(S^{n-1})$ is called a multiplier transformation [16] if there exist real numbers $c_k$, the multipliers of Φ, such that, for every $k\in \mathbb{N}$,

$${\pi }_k\mathrm{\Phi }\mu =c_k{\pi }_k\mu ,\mathrm{\forall }\mu \in \mathcal{D}.$$

(2.21)

From the Funk-Hecke theorem and the fact that the spherical convolution of zonal measures is commutative, it follows that, for $\mu \in \mathcal{M}(S^{n-1},\hat{e})$, the map ${\mathrm{\Phi }}_{\mu }:\mathcal{M}(S^{n-1})\to \mathcal{M}(S^{n-1})$, defined by ${\mathrm{\Phi }}_{\mu }=\nu \ast \mu$, is a multiplier transformation. The multipliers of this convolution operator are just the Legendre coefficients of the measure μ.

3 $L_p$ Blaschke-Minkowski homomorphisms and convolutions

The $L_p$ Minkowski valuation was introduced by Ludwig [11]. A function $\mathrm{\Psi }:{\mathcal{K}}_0^n\to {\mathcal{K}}_0^n$ is called an $L_p$ Minkowski valuation if

$$\mathrm{\Psi }(K\cup L){+}_p\mathrm{\Psi }(K\cap L)=\mathrm{\Psi }K{+}_p\mathrm{\Psi }L,$$

(3.1)

whenever $K,L,K\cup L\in {\mathcal{K}}_0^n$, and here ‘${+}_p$’ is $L_p$ Minkowski addition.

Definition 3.1 A map ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ satisfying the following properties (a), (b) and (c) is called an $L_p$ Blaschke-Minkowski homomorphism.

• (a)
  $${\mathrm{\Phi }}_p$$

  is continuous with respect to Hausdorff metric.

• (b)
  $${\mathrm{\Phi }}_p(K{\mathrm{\#}}_{p}L)={\mathrm{\Phi }}_{p}K{+}_p{\mathrm{\Phi }}_{p}L$$

  for all $K,L\in {\mathcal{K}}_e^n$.

• (c)
  $${\mathrm{\Phi }}_p$$

  is $SO(n)$ equivariant, i.e., ${\mathrm{\Phi }}_p(\vartheta K)=\vartheta {\mathrm{\Phi }}_{p}K$ for all $\vartheta \in SO(n)$ and all $K\in {\mathcal{K}}_e^n$.

It is easy to verify that an $L_p$ Blaschke-Minkowski homomorphism is an $L_p$ Minkowski valuation.

In order to prove our results, we need to quote some lemmas. We call a map $\mathrm{\Phi }:\mathcal{M}(S^{n-1})\to \mathcal{C}(S^{n-1})$ monotone, if non-negative measures are mapped to non-negative functions.

Lemma 3.1 A map $\mathrm{\Phi }:\mathcal{M}(S^{n-1})\to \mathcal{C}(S^{n-1})$ is a monotone, linear map that is intertwines rotations if and only if there is a function $f\in \mathcal{C}(S^{n-1},\hat{e})$, such that

$$\mathrm{\Phi }\mu =f\ast \mu .$$

(3.2)

Proof From the definition of spherical convolution and (2.15), it follows that mapping of form (3.2) has the desired properties. This proves the sufficiency.

Next, we prove the necessity.

Let Φ be monotone, linear and intertwines rotations. Consider the map $\varphi :\mathcal{M}(S^{n-1})\to \mathbb{R}$, $\mu \to \mathrm{\Phi }\mu (\hat{e})$. By the properties of Φ, the functional ϕ is positive and linear on $\mathcal{M}(S^{n-1})$, thus, by the Riesz representation theorem, there is a function $f\in {\mathcal{M}}_{+}(S^{n-1})$ such that

$$\varphi (\mu )={\int }_{S^{n-1}}f(u)d\mu (u).$$

Since ϕ is $SO(n-1)$ invariant, the function f is zonal. Thus, we have for $\eta \in SO(n)$

$$\mathrm{\Phi }\mu (\eta \hat{e})=\mathrm{\Phi }\left({\eta }^{-1}\mu \right)(\hat{e})=\varphi \left({\eta }^{-1}\mu \right)={\int }_{S^{n-1}}f(\eta u)d\mu (u).$$

Lemma 3.1 follows now from (2.14). □

Proof of Theorem 1.1 Suppose that a map ${\mathrm{\Phi }}_p:{\mathcal{K}}_0^n\to {\mathcal{K}}_0^n$ satisfies $h{({\mathrm{\Phi }}_{p}K,\cdot )}^p=S_p(K,\cdot )\ast g$, where $g\in \mathcal{C}(S^{n-1},\hat{e})$ is a nonnegative measure. The continuity of ${\mathrm{\Phi }}_p$ follows from the fact that the support function $h(K,\cdot )$ is continuous with respect to Hausdorff metric. From (2.9) and (2.1), for $\vartheta \in SO(n)$, we obtain

$$h{({\mathrm{\Phi }}_p\vartheta K,\cdot )}^p=S_p(\vartheta K,\cdot )\ast g=S_p(K,{\vartheta }^{-1}\cdot )\ast g=h{({\mathrm{\Phi }}_{p}K,{\vartheta }^{-1}\cdot )}^p=h{(\vartheta {\mathrm{\Phi }}_{p}K,\cdot )}^p.$$

Taking $K=L$ in (1.4), we have

$$h{({\mathrm{\Phi }}_{p}L,\cdot )}^p=S_p(L,\cdot )\ast g.$$

(3.3)

Combining with (2.2), (1.4) and (3.3), we obtain

$$\begin{array}{rl}h{({\mathrm{\Phi }}_{p}K{+}_p{\mathrm{\Phi }}_{p}L,\cdot )}^p& =h{({\mathrm{\Phi }}_{p}K,\cdot )}^p+h{({\mathrm{\Phi }}_{p}L,\cdot )}^p\\ =S_p(K,\cdot )\ast g+S_p(L,\cdot )\ast g\\ =(S_p(K,\cdot )+S_p(L,\cdot ))\ast g\\ =S_p(K{\mathrm{\#}}_{p}L,\cdot )\ast g\\ =h{({\mathrm{\Phi }}_p(K{\mathrm{\#}}_{p}L),\cdot )}^p.\end{array}$$

(3.4)

Thus maps of the form of (1.4) are $L_p$ Blaschke-Minkowski homomorphisms (satisfy the properties (a), (b) and (c) from Definition 3.1). Thus, we have to show that for every such operator ${\mathrm{\Phi }}_p$, there is a function $g\in \mathcal{C}(S^{n-1},\hat{e})$ such that (1.4) holds.

Since every positive continuous even measure on $S^{n-1}$ can be the $L_p$ surface area measure of some convex body, the set $\{S_p(K,\cdot )-S_p(L,\cdot ),K,L\in {\mathcal{K}}_e^n\}$ coincides with ${\mathcal{M}}_e(S^{n-1})$. The operator $\overline{\mathrm{\Phi }}:\mathcal{M}(S^{n-1})\to \mathcal{C}(S^{n-1})$ is defined by

$$\overline{\mathrm{\Phi }}{\mu }_1=h{({\mathrm{\Phi }}_p{K}_1,\cdot )}^p-h{({\mathrm{\Phi }}_p{K}_2,\cdot )}^p,$$

(3.5)

where ${\mu }_1=S_p(K_1,\cdot )-S_p(K_2,\cdot )$.

The operator $\overline{\mathrm{\Phi }}$ for ${\mu }_2=S_p(L_1,\cdot )-S_p(L_2,\cdot )$ immediately yields:

$$\overline{\mathrm{\Phi }}{\mu }_2=h{({\mathrm{\Phi }}_p{L}_1,\cdot )}^p-h{({\mathrm{\Phi }}_p{L}_2,\cdot )}^p.$$

(3.6)

Combining with (3.5), (3.6), (2.2) and (3.4), we obtain

$$\begin{array}{rl}\overline{\mathrm{\Phi }}{\mu }_1+\overline{\mathrm{\Phi }}{\mu }_2& =h{({\mathrm{\Phi }}_p{K}_1,\cdot )}^p-h{({\mathrm{\Phi }}_p{K}_2,\cdot )}^p+h{({\mathrm{\Phi }}_p{L}_1,\cdot )}^p-h{({\mathrm{\Phi }}_p{L}_2,\cdot )}^p\\ =h{({\mathrm{\Phi }}_p{K}_1{+}_p{\mathrm{\Phi }}_p{L}_1,\cdot )}^p-h{({\mathrm{\Phi }}_p{K}_2{+}_p{\mathrm{\Phi }}_p{L}_2,\cdot )}^p\\ =h{({\mathrm{\Phi }}_p(K_1{\mathrm{\#}}_p{L}_1),\cdot )}^p-h{({\mathrm{\Phi }}_p(K_2{\mathrm{\#}}_p{L}_2),\cdot )}^p\\ =\overline{\mathrm{\Phi }}(S_p(K_1{\mathrm{\#}}_p{L}_1,\cdot )-S_p(K_2{\mathrm{\#}}_p{L}_2,\cdot ))\\ =\overline{\mathrm{\Phi }}(S_p(K_1,\cdot )+S_p(L_1,\cdot )-S_p(K_2,\cdot )-S_p(L_2,\cdot ))\\ =\overline{\mathrm{\Phi }}({\mu }_1+{\mu }_2).\end{array}$$

So, the operator $\overline{\mathrm{\Phi }}$ is linear.

Noting that ${\mathrm{\Phi }}_p$ is an $L_p$ Minkowski homomorphism and $S_p(\vartheta K,\cdot )=\vartheta S_p(K,\cdot )$, we obtain that the operator $\overline{\mathrm{\Phi }}$ is $SO(n)$ equivariant.

Since the cone of the $L_p$ surface area measures of origin symmetric convex bodies is invariant under $\overline{\mathrm{\Phi }}$, it is also monotone. Hence, by Lemma 3.1, there is a non-negative function $g\in \mathcal{C}(S^{n-1},\hat{e})$ such that $\overline{\mathrm{\Phi }}\mu =\mu \ast g$. The statement now follows from

$$\overline{\mathrm{\Phi }}{S}_p(K,\cdot )=S_p(K,\cdot )\ast g=h{({\mathrm{\Phi }}_{p}K,\cdot )}^p.$$

Hence, it is to complete the proof. □

Lutwak, Yang and Zhang first introduced the notion of $L_p$-projection body (see [24]). Let ${\mathrm{\Pi }}_{p}K$, $p\ge 1$ denote the compact convex symmetric set whose support function is given by

$$h{({\mathrm{\Pi }}_{p}K,\theta )}^p=\frac{1}{n{\omega }_n{c}_{n-2,p}}{S}_p(K,\cdot )\ast {|〈\theta ,\cdot 〉|}^p,$$

(3.7)

where

$$c_{n,p}=\frac{{\omega }_{n+p}}{{\omega }_2{\omega }_n{\omega }_{p-1}}.$$

Obviously, ${\mathrm{\Pi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ is an $L_p$ Blaschke-Minkowski homomorphism.

Lemma 3.2 [23]

If $\mu ,\nu \in \mathcal{M}(S^{n-1})$ and $f\in \mathcal{C}(S^{n-1})$, then

$$〈\mu \ast \nu ,f〉=〈\mu ,f\ast \nu 〉.$$

Theorem 3.3 If ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ is an $L_p$ Blaschke-Minkowski homomorphism, then for $K,L\in {\mathcal{K}}_e^n$,

$$V_p(K,{\mathrm{\Phi }}_{p}L)=V_p(L,{\mathrm{\Phi }}_{p}K).$$

(3.8)

Proof Let $g\in \mathcal{C}(S^{n-1},\hat{e})$ be the generating function of ${\mathrm{\Phi }}_p$. Using (2.3), Theorem 1.1 and Lemma 3.2, it follows that

$$\begin{array}{rl}n{V}_p(K,{\mathrm{\Phi }}_{p}L)& =〈h{({\mathrm{\Phi }}_{p}L,\cdot )}^p,S_p(K,\cdot )〉\\ =〈S_p(L,\cdot )\ast g,S_p(K,\cdot )〉\\ =〈S_p(L,\cdot ),S_p(K,\cdot )\ast g〉\\ =〈S_p(L,\cdot ),h{({\mathrm{\Phi }}_{p}K,\cdot )}^p〉\\ =n{V}_p(L,{\mathrm{\Phi }}_{p}K).\end{array}$$

(3.9)

 □

Using Theorem 1.1 and the fact that spherical convolution operators are multiplier transformations, one obtains the following lemma.

Lemma 3.4 If ${\mathrm{\Phi }}_p$ is an $L_p$ Blaschke-Minkowski homomorphism, which is generated by the zonal function g, then for every origin symmetric convex body $K\in {\mathcal{K}}_e^n$,

$${\pi }_{k}h{({\mathrm{\Phi }}_{p}K,\cdot )}^p=g_k{\pi }_k{S}_p(K,\cdot ),k\in \mathbb{N},$$

(3.10)

where the numbers $g_k$ are the Legendre coefficients of g, i.e., $g_k=〈g,\mathrm{\Lambda }{P}_k^n〉$.

Proof By (2.18) and Theorem 1.1, we have

$${\pi }_{k}h{({\mathrm{\Phi }}_{p}K,\cdot )}^p=N(n,k)(S_p(K,\cdot )\ast g\ast \mathrm{\Lambda }{P}_k^n).$$

Since spherical convolution is associative and g is zonal, we obtain from (2.18):

$${\pi }_{k}h{({\mathrm{\Phi }}_{p}K,\cdot )}^p=g_{k}N(n,k)(S_p(K,\cdot )\ast \mathrm{\Lambda }{P}_k^n)=g_k{\pi }_k{S}_p(K,\cdot ).$$

 □

Definition 3.2 If ${\mathrm{\Phi }}_p$ is an $L_p$ Blaschke-Minkowski homomorphism, generated by the zonal function g, then we call the subset ${\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$ of ${\mathcal{K}}_e^n$, defined by

$${\mathcal{K}}_e^n({\mathrm{\Phi }}_p)=\{K\in {\mathcal{K}}_e^n:{\pi }_k{S}_p(K,\cdot )=0\text{ if }{g}_k=0\},$$

the injectivity set of ${\mathrm{\Phi }}_p$.

It is easy to verify that for every $L_p$ Blaschke-Minkowski homomorphism, the set is a nonempty rotation and dilatation invariant subset of which is closed under $L_p$ Blaschke addition.

Definition 3.3 An origin-symmetric convex body $K\in {\mathcal{K}}_e^n$ p-polynomial if $h{(K,\cdot )}^p\in {\mathcal{H}}^n$.

Clearly, the set of p-polynomial convex bodies is dense in ${\mathcal{K}}_e^n$.

Let $p>1$ and $p\ne n$ where p is not an even integer. The size of range, ${\mathrm{\Phi }}_p({\mathcal{K}}_e^n)$, of the $L_p$ Blaschke-Minkowski homomorphism ${\mathrm{\Phi }}_p$ will be critical. The set of origin-symmetric convex bodies whose support functions are elements of the vector space

$$span\{{(h{({\mathrm{\Phi }}_{p}K,\cdot )}^p-h{({\mathrm{\Phi }}_{p}L,\cdot )}^p)}^{\frac{1}{p}}:K,L\in {\mathcal{K}}_e^n\}$$

(3.11)

is a large subset of ${\mathcal{K}}_e^n$, provided the injectivity set ${\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$ is not too small.

Theorem 3.5 Let $p>1$ and $p\ne n$ where p is not an even integer. If ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ is an $L_p$ Blaschke-Minkowski homomorphism such that ${\mathcal{K}}_e^n\subseteq {\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$, then for every p-polynomial convex body $K\in {\mathcal{K}}_e^n$, there exist origin-symmetry convex bodies $K_1,K_2\in {\mathcal{K}}_e^n$ such that

$$K{+}_p{\mathrm{\Phi }}_p{K}_1={\mathrm{\Phi }}_p{K}_2.$$

(3.12)

Proof Let $K\in {\mathcal{K}}_e^n$ be a p-polynomial convex body. From Definition 3.3, we have

$$h{(K,\cdot )}^p=\sum _{k=0}^m{\pi }_{k}h{(K,\cdot )}^p.$$

(3.13)

For $K\in {\mathcal{K}}_e^n$ and the properties of the orthogonal projection of f on the space ${\mathcal{H}}_k^n$, we have ${\pi }_{k}h{(K,\cdot )}^p=0$ for all odd $k\in \mathbb{N}$. Let $g\in \mathcal{C}(S^{n-1},\hat{e})$ denote the generating function of Φ and let $g_k$ denote the Legendre coefficients of g. From ${\mathcal{K}}_e^n\subseteq {\mathcal{K}}_e^n(\mathrm{\Phi })$ and Definition 3.2, it follows that $g_k\ne 0$ for every even $k\in \mathbb{N}$. We define

$$f:=\sum _{k=0}^m{c}_k{\pi }_{k}h{(K,\cdot )}^p,$$

(3.14)

where $c_k=0$ for odd and $c_k=g_k^{-1}$ if k is even. Since f is an even continuous function on $S^{n-1}$ and spherical convolution operators are multiplier transformations, we have

$$f\ast g=\sum _{k=0}^m{c}_k{g}_k{\pi }_{k}h{(K,\cdot )}^p=\sum _{k=0}^m{\pi }_{k}h{(K,\cdot )}^p=h{(K,\cdot )}^p.$$

(3.15)

Denote by $f^{+}$ and $f^{-}$ the positive and negative parts of f and let $K_1$ and $K_2$ be the convex bodies such that $S_p(K_1,\cdot )=f^{-}$ and $S_p(K_2,\cdot )=f^{+}$. By Theorem 1.1 and (2.2), it follows that

$$K{+}_p{\mathrm{\Phi }}_p{K}_1={\mathrm{\Phi }}_p{K}_2.$$

 □

4 The Shephard-type problem

Let ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ denote a nontrivial $L_p$ Blaschke-Minkowski homomorphism, i.e., ${\mathrm{\Phi }}_p$ is continuous and $SO(n)$ equivariant map satisfying ${\mathrm{\Phi }}_p(K{\mathrm{\#}}_{p}L)={\mathrm{\Phi }}_{p}K{+}_p{\mathrm{\Phi }}_{p}L$ and ${\mathrm{\Phi }}_p$ does not map every origin-symmetric convex body to the origin. In this section, we study the Shephard-type problem for $L_p$ Blaschke-Minkowski homomorphisms.

Problem 4.1 Let $p>1$, $p\ne n$ and ${\mathrm{\Phi }}_p:{\mathcal{K}}_0^n\to {\mathcal{K}}_e^n$ be an $L_p$ Blaschke-Minkowski homomorphism. Is there the implication:

If $0<p<n$, then

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\Rightarrow V(K)\le V(L)?$$

(4.1)

If $p>n$, then

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\Rightarrow V(K)\ge V(L)?$$

(4.2)

Proof of Theorem 1.2 For $L\in {\mathrm{\Phi }}_p{\mathcal{K}}_e^n$ and p is not an even integer, there exists an origin-symmetric convex body $L_0$ such that $L={\mathrm{\Phi }}_p{L}_0$. Using Theorem 3.3 and the fact that the $L_p$ mixed volume $V_p$ is monotone with respect to set inclusion, it follows that

$$V_p(K,L)=V_p(K,{\mathrm{\Phi }}_p{L}_0)=V_p(L_0,{\mathrm{\Phi }}_{p}K)\le V_p(L_0,{\mathrm{\Phi }}_{p}L)=V_p(L,{\mathrm{\Phi }}_p{L}_0)=V(L).$$

Applying the $L_p$ Minkowski inequality (2.6), we thus obtain that, if $1<p<n$, then

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

and if $p>n$, then

$$V(K)\ge V(L),$$

with equality if and only if K and L are dilates. □

An immediate consequence of Theorem 1.2 is the following.

Theorem 4.1 Let $p>1$, $p\ne n$, where p is not an even integer and ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ is an $L_p$ Blaschke-Minkowski homomorphism. If $K,L\in {\mathrm{\Phi }}_p{\mathcal{K}}_e^n$, then

$${\mathrm{\Phi }}_{p}K={\mathrm{\Phi }}_{p}L\iff K=L.$$

(4.3)

Since the $L_p$ projection body operator ${\mathrm{\Pi }}_p$ is just an $L_p$ Blaschke-Minkowski homomorphism, the $L_p$ Aleksandrov’s projection theorem is a direct corollary of Theorem 4.1.

Corollary 4.2 [25]

Let $p>1$, $p\ne n$, where p is not an even integer, and K and L are both $L_p$ projection bodies in ${\mathbb{R}}^n$. Then

$${\mathrm{\Pi }}_{p}K={\mathrm{\Pi }}_{p}L\iff K=L.$$

Our next result shows that if the injectivity set ${\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$ does not exhaust all of ${\mathcal{K}}_e^n$, in general the answer to Problem 4.1 is negative.

Theorem 4.3 Let $1<p<n$ where p is not an even integer. If ${\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$ does not coincide with ${\mathcal{K}}_e^n$, then there exist origin-symmetric convex bodies $K,L\in {\mathcal{K}}_e^n$, such that

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L,$$

but

$$V(K)>V(L).$$

Proof Let $g\in \mathcal{C}(S^{n-1},\hat{e})$ be the generating function of ${\mathrm{\Phi }}_p$ and let $g_k$ denote its Legendre coefficients. Since ${\mathcal{K}}_e^n({\mathrm{\Phi }}_p)\ne {\mathcal{K}}_e^n$ and ${\mathrm{\Phi }}_p$ is nontrivial, there exists, by Definition 3.2, an integer $k\in \mathbb{N}$, such that $g_k=0$ and $k\ge 1$. We can choose $\alpha >0$ such that the function $f(u)=1+\alpha P_k^n(u\cdot \hat{e})$, $u\in S^{n-1}$, is positive. According to Theorem C, there exists an origin-symmetric convex body $L\in {\mathcal{K}}_e^n$ with $S_p(L,\cdot )=f$.

Since ${\pi }_k{S}_p(L,\cdot )={\pi }_k(1+\alpha P_k^n(u\cdot \hat{e}))\ne 0$, from Definition 3.2 we have that $L\notin {\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$.

From (2.20) and the properties of the orthogonal projection on the space ${\mathcal{H}}_k^n$, we have that

$$n{V}_p(L,B)={\pi }_0{S}_p(L,\cdot )=1.$$

(4.4)

Using the fact that: For $1<p<n$ where p is not an even integer, an origin-symmetric convex body $L\in {\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$ is uniquely determined by its image ${\mathrm{\Phi }}_{p}L$, we obtain that ${\mathrm{\Phi }}_{p}L={\mathrm{\Phi }}_{p}K$, where K denotes the Euclidean ball centered at the origin with $L_p$ surface area $S_p(K)=1$. Noting that L is just a perturb body of K, we use (4.4) and (2.6) to conclude

$$V{(K)}^{n-p}=\frac{1}{n^{n}V{(B)}^p}>V{(L)}^{n-p}.$$

 □

Theorem 4.4 Suppose $1<p<n$ where p is not an even integer and ${\mathcal{K}}_e^n\subseteq {\mathcal{K}}_e^n({\mathrm{\Phi }}_p)$. If $K\in {\mathcal{K}}_e^n$ is a p-polynomial convex body which has p-positive curvature function, then if $K\notin {\mathrm{\Phi }}_p{\mathcal{K}}_e^n$, there exists an origin-symmetric convex body $L\in {\mathcal{K}}_e^n$, such that

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L,$$

but

$$V(K)>V(L).$$

Proof Let $g\in \mathcal{C}(S^{n-1},\hat{e})$ be the generating function of ${\mathrm{\Phi }}_p$. Since $K\in {\mathcal{K}}_e^n$ is p-polynomial, it follows from the proof of Theorem 3.5 that there exists an even function $f\in {\mathcal{H}}^n$ such that

$$h{(K,\cdot )}^p=f\ast g.$$

(4.5)

The function must assume negative values, otherwise, by Theorem 1.1 we have $K={\mathrm{\Phi }}_p{K}_0$, where $K_0$ is the convex body with $S_p(K_0,\cdot )=f$. Let $F\in \mathcal{C}(S^{n-1})$ be a non-constant even function, such that: $F(u)\ge 0$ if $f(u)<0$, and $F(u)=0$ if $f(u)\ge 0$. By suitable approximation of the function F with spherical harmonics, we can find a nonnegative even function $G\in {\mathcal{H}}^n$ and an even function $H\in {\mathcal{H}}^n$ such that

$$〈f,G〉<0,\text{and}G=H\ast g.$$

(4.6)

Since K is a p-polynomial and has p-positive curvature, the $L_p$ surface area measure of K has a positive density $S_p(K,\cdot )$. Thus, we can choose $\alpha >0$ such that

$$S_p(K,\cdot )+\alpha H>0.$$

By Theorem C, there exists an origin-symmetric convex body L such that

$$S_p(L,\cdot )=S_p(K,\cdot )+\alpha H.$$

(4.7)

From (4.6) and Theorem 1.1, we see that $h{({\mathrm{\Phi }}_{p}L,\cdot )}^p=h{({\mathrm{\Phi }}_{p}K,\cdot )}^p+\alpha G$.

Since $G\ge 0$, it follows that

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L.$$

(4.8)

Applying with (2.3), (4.5), (4.7), (2.10) and (4.6), we obtain

$$\begin{array}{rl}n(V_p(K,L)-V(K))& =〈h{(K,\cdot )}^p,S_p(L,\cdot )-S_p(K,\cdot )〉\\ =〈h{(K,\cdot )}^p,\alpha H〉\\ =\alpha 〈f\ast g,H〉\\ =\alpha 〈f,H\ast g〉\\ =\alpha 〈f,G〉<0.\end{array}$$

(4.9)

To complete the proof, we can use (2.6) to conclude

$$V(K)>V(L).$$

 □

In particular, we replace ${\mathrm{\Phi }}_p$ by ${\mathrm{\Pi }}_p$ to Theorem 1.2, we have the following corollary, which was proved by Ryabogin and Zvavitch.

Corollary 4.5 [25]

Let K and L be origin-symmetric convex bodies and $1\le p<n$ where p is not an even integer. If L belongs to the class of $L_p$ projection bodies, then

$${\mathrm{\Pi }}_{p}K\subseteq {\mathrm{\Pi }}_{p}L\Rightarrow V(K)\le V(L).$$