© Wang; licensee Springer 2013
Received: 25 March 2012
Accepted: 19 March 2013
Published: 2 April 2013
In this paper, we introduce the concept of 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 Blaschke-Minkowski homomorphisms.
The theory of real valued valuations is at the center of convex geometry. Blaschke started a systematic investigation in the 1930s, and then Hadwiger  focused on classifying valuations on compact convex sets in and obtained the famous Hadwiger’s characterization theorem. Schneider  obtained first results on convex body valued valuations with Minkowski addition in 1970s. The survey  and the book  are an excellent source for the classical theory of valuations. Some more recent results can see [1, 5–20].
whenever , and here + is the Minkowski addition.
A map is called a Blaschke-Minkowski homomorphism if it is continuous, equivariant and satisfies , where # denotes the Blaschke addition, i.e., .
Obviously, a Blaschke-Minkowski homomorphism is a continuous Minkowski valuation which is equivariant and -homogeneous. Schuster introduced Blaschke-Minkowski homomorphisms and studied the Busemann-Petty type problem for them.
Theorem A 
Theorem B 
and if and only if .
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 . She started systematic studies and established complete classifications of convex and star body valued valuations with respect to Minkowski addition and radial which are compatible with the action of the group . Based on these results, in this article we study Blaschke-Minkowski homomorphisms which are continuous, -homogeneous and equivariant.
and , if and only if .
2 Notation and background material
where is the inverse of ϑ.
where ‘ ⋅ ’ in denotes the Firey scalar multiplication, i.e., .
if , equality holds if and only if K and L are dilates; if , equality holds if and only if K and L are homothetic.
The Minkowski problem asks for necessary and sufficient conditions for a Borel measure μ on to be the surface area measure of a convex body. Lutwak  gave a weak solution to the Minkowski problem as follows.
where is the image measure of under the rotation ϑ. Obviously, is just .
Some basic notions on spherical harmonics will be required. The article by Grinberg and Zhang  and the article by Schuster  are excellent general references on spherical harmonics. As usual, and will be equipped with the invariant probability measures. Let , be the spaces of continuous functions on and with uniform topology and , their dual spaces of signed finite Borel measures with weak∗ topology. The group acts on these spaces by left translation, i.e., for and , we have , , and ϑμ is the image measure of μ under the rotation ϑ.
The sphere is identified with the homogeneous space , where denotes the subgroup of rotations leaving the pole of fixed. The projection from onto is . Functions on can be identified with right -invariant functions on , by , for . In fact, is isomorphic to the subspace of right -invariant functions in .
We denote by the finite dimensional vector space of spherical harmonics of dimension n and order k, and let be the dimension of . The space of all finite sums of spherical harmonics of dimension n is denoted by . The spaces are pairwise orthogonal with respect to the usual inner product on . Clearly, is invariant with respect to rotations.
A zonal measure is defined by its so-called Legendre coefficients . Using for every and the fact that spherical convolution of zonal measures is commutative, we have the Funk-Hecke theorem: If and , then .
From the Funk-Hecke theorem and the fact that the spherical convolution of zonal measures is commutative, it follows that, for , the map , defined by , is a multiplier transformation. The multipliers of this convolution operator are just the Legendre coefficients of the measure μ.
3 Blaschke-Minkowski homomorphisms and convolutions
whenever , and here ‘’ is Minkowski addition.
- (a)is continuous with respect to Hausdorff metric.
- (b)for all .
- (c)is equivariant, i.e., for all and all .
It is easy to verify that an Blaschke-Minkowski homomorphism is an Minkowski valuation.
In order to prove our results, we need to quote some lemmas. We call a map monotone, if non-negative measures are mapped to non-negative functions.
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.
Lemma 3.1 follows now from (2.14). □
Thus maps of the form of (1.4) are Blaschke-Minkowski homomorphisms (satisfy the properties (a), (b) and (c) from Definition 3.1). Thus, we have to show that for every such operator , there is a function such that (1.4) holds.
So, the operator is linear.
Noting that is an Minkowski homomorphism and , we obtain that the operator is equivariant.
Hence, it is to complete the proof. □
Obviously, is an Blaschke-Minkowski homomorphism.
Lemma 3.2 
Using Theorem 1.1 and the fact that spherical convolution operators are multiplier transformations, one obtains the following lemma.
where the numbers are the Legendre coefficients of g, i.e., .
the injectivity set of .
It is easy to verify that for every Blaschke-Minkowski homomorphism, the set is a nonempty rotation and dilatation invariant subset of which is closed under Blaschke addition.
Definition 3.3 An origin-symmetric convex body p-polynomial if .
Clearly, the set of p-polynomial convex bodies is dense in .
is a large subset of , provided the injectivity set is not too small.
4 The Shephard-type problem
Let denote a nontrivial Blaschke-Minkowski homomorphism, i.e., is continuous and equivariant map satisfying and does not map every origin-symmetric convex body to the origin. In this section, we study the Shephard-type problem for Blaschke-Minkowski homomorphisms.
Problem 4.1 Let , and be an Blaschke-Minkowski homomorphism. Is there the implication:
with equality if and only if K and L are dilates. □
An immediate consequence of Theorem 1.2 is the following.
Since the projection body operator is just an Blaschke-Minkowski homomorphism, the Aleksandrov’s projection theorem is a direct corollary of Theorem 4.1.
Corollary 4.2 
Our next result shows that if the injectivity set does not exhaust all of , in general the answer to Problem 4.1 is negative.
Proof Let be the generating function of and let denote its Legendre coefficients. Since and is nontrivial, there exists, by Definition 3.2, an integer , such that and . We can choose such that the function , , is positive. According to Theorem C, there exists an origin-symmetric convex body with .
Since , from Definition 3.2 we have that .
From (4.6) and Theorem 1.1, we see that .
In particular, we replace by to Theorem 1.2, we have the following corollary, which was proved by Ryabogin and Zvavitch.
Corollary 4.5 
A project supported by Scientific Research Fund of Hunan Provincial Education Department (11C0542).
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