Journal of Inequalities and Applications volume 2013, Article number: 140 (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].
An operator is called a Minkowski valuation if
whenever , and here + is the Minkowski addition.
A Minkowski valuation Z is called equivariant, if for all and all ,
A Minkowski valuation Z is called homogeneity of degree p, if for all and all ,
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 
If be a Blaschke-Minkowski homomorphism, then there is a weakly positive , unique up to a linear function, such that
Theorem B 
Let be a Blaschke-Minkowski homomorphism. If and , then
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.
Theorem 1.1 Let and . If be an Blaschke-Minkowski homomorphism, then there is a nonnegative function , such that
Theorem 1.2 Let and p is not an even integer, and let be an Blaschke-Minkowski homomorphism. If and , then
If and p is not an even integer, then
and , if and only if .
2 Notation and background material
Let denote the set of convex bodies containing the origin in their interiors, and let denote origin-symmetric convex bodies. In this paper, we restrict the dimension of to . A convex body is uniquely determined by its support function, . From the definition of , it follows immediately that for and ,
where is the inverse of ϑ.
For , , and , the Minkowski addition is defined by (see )
where ‘ ⋅ ’ in denotes the Firey scalar multiplication, i.e., .
If , then for , the mixed volume, , of K and L is defined by (see )
Corresponding to each , there is a positive Borel measure, , on such that (see )
for each . The measure is just the surface area measure of K, which is absolutely continuous with respect to classical surface area measure , and has a Radon-Nikodym derivative
A convex body is said to have a p-curvature function (see ) , if its surface area measure is absolutely continuous with respect to spherical Lebesgue measure S and the Radon-Nikodym derivative
From the formula (2.3), it follows immediately that for each ,
The Minkowski inequality for the mixed volume states that (see ): For , if , then
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.
Theorem C If μ is an even position Borel measure on , which is not concentrated on any great subsphere, then for any and , there exists a unique origin-symmetric convex bodies , such that
From (2.4), for , we have
Noting the fact for and (2.1), one can obtain
where is the image measure of under the rotation ϑ. Obviously, is just .
The Blaschke addition of is the convex body with
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 .
The convolution of a measure and a function is defined by
The canonical pairing of and is defined by
A function is called zonal, if for every . Zonal functions depend only on the value . The set of continuous zonal functions on will be denoted by and the definition of is analogous. A map is defined by
The map Λ is also an isomorphism between functions on and zonal functions on . If , and , then
If , for each and every , then
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.
Let denote the Legendre polynomial of dimension n and order k. The zonal function is up to a multiplicative constant the unique zonal spherical harmonic in . In each space we choose an orthonormal basis . The collection forms a complete orthogonal system in . In particular, for every , the series
converges to f in the -norm, where is the orthogonal projection of f on the space . Using well-known properties of the Legendre polynomials, it is not hard to show that
This leads to the spherical expansion of a measure ,
where is defined by
From , and , , we obtain, for , the following special cases of (2.18):
Let denote the volume of the Euclidean unit ball B. By (2.3) and (2.19), for every convex body , it follows that
A measure is uniquely determined by its series expansion (2.19). Using the fact that is (essentially) the unique zonal function in , a simple calculation shows that for , formula (2.18) becomes
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 .
A map is called a multiplier transformation  if there exist real numbers , the multipliers of Φ, such that, for every ,
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
The Minkowski valuation was introduced by Ludwig . A function is called an Minkowski valuation if
whenever , and here ‘’ is Minkowski addition.
Definition 3.1 A map satisfying the following properties (a), (b) and (c) is called an Blaschke-Minkowski homomorphism.
is continuous with respect to Hausdorff metric.
for all .
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.
Lemma 3.1 A map is a monotone, linear map that is intertwines rotations if and only if there is a function , such that
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 , . By the properties of Φ, the functional ϕ is positive and linear on , thus, by the Riesz representation theorem, there is a function such that
Since ϕ is invariant, the function f is zonal. Thus, we have for
Lemma 3.1 follows now from (2.14). □
Proof of Theorem 1.1 Suppose that a map satisfies , where is a nonnegative measure. The continuity of follows from the fact that the support function is continuous with respect to Hausdorff metric. From (2.9) and (2.1), for , we obtain
Taking in (1.4), we have
Combining with (2.2), (1.4) and (3.3), we obtain
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.
Since every positive continuous even measure on can be the surface area measure of some convex body, the set coincides with . The operator is defined by
The operator for immediately yields:
Combining with (3.5), (3.6), (2.2) and (3.4), we obtain
So, the operator is linear.
Noting that is an Minkowski homomorphism and , we obtain that the operator is equivariant.
Since the cone of the surface area measures of origin symmetric convex bodies is invariant under , it is also monotone. Hence, by Lemma 3.1, there is a non-negative function such that . The statement now follows from
Hence, it is to complete the proof. □
Lutwak, Yang and Zhang first introduced the notion of -projection body (see ). Let , denote the compact convex symmetric set whose support function is given by
Obviously, is an Blaschke-Minkowski homomorphism.
Lemma 3.2 
If and , then
Theorem 3.3 If is an Blaschke-Minkowski homomorphism, then for ,
Proof Let be the generating function of . Using (2.3), Theorem 1.1 and Lemma 3.2, it follows that
Using Theorem 1.1 and the fact that spherical convolution operators are multiplier transformations, one obtains the following lemma.
Lemma 3.4 If is an Blaschke-Minkowski homomorphism, which is generated by the zonal function g, then for every origin symmetric convex body ,
where the numbers are the Legendre coefficients of g, i.e., .
Proof By (2.18) and Theorem 1.1, we have
Since spherical convolution is associative and g is zonal, we obtain from (2.18):
Definition 3.2 If is an Blaschke-Minkowski homomorphism, generated by the zonal function g, then we call the subset of , defined by
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 .
Let and where p is not an even integer. The size of range, , of the Blaschke-Minkowski homomorphism will be critical. The set of origin-symmetric convex bodies whose support functions are elements of the vector space
is a large subset of , provided the injectivity set is not too small.
Theorem 3.5 Let and where p is not an even integer. If is an Blaschke-Minkowski homomorphism such that , then for every p-polynomial convex body , there exist origin-symmetry convex bodies such that
Proof Let be a p-polynomial convex body. From Definition 3.3, we have
For and the properties of the orthogonal projection of f on the space , we have for all odd . Let denote the generating function of Φ and let denote the Legendre coefficients of g. From and Definition 3.2, it follows that for every even . We define
where for odd and if k is even. Since f is an even continuous function on and spherical convolution operators are multiplier transformations, we have
Denote by and the positive and negative parts of f and let and be the convex bodies such that and . By Theorem 1.1 and (2.2), it follows that
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:
If , then
If , then
Proof of Theorem 1.2 For and p is not an even integer, there exists an origin-symmetric convex body such that . Using Theorem 3.3 and the fact that the mixed volume is monotone with respect to set inclusion, it follows that
Applying the Minkowski inequality (2.6), we thus obtain that, if , then
and if , then
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 , , where p is not an even integer and is an Blaschke-Minkowski homomorphism. If , then
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 
Let , , where p is not an even integer, and K and L are both projection bodies in . Then
Our next result shows that if the injectivity set does not exhaust all of , in general the answer to Problem 4.1 is negative.
Theorem 4.3 Let where p is not an even integer. If does not coincide with , then there exist origin-symmetric convex bodies , such that
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 (2.20) and the properties of the orthogonal projection on the space , we have that
Using the fact that: For where p is not an even integer, an origin-symmetric convex body is uniquely determined by its image , we obtain that , where K denotes the Euclidean ball centered at the origin with surface area . Noting that L is just a perturb body of K, we use (4.4) and (2.6) to conclude
Theorem 4.4 Suppose where p is not an even integer and . If is a p-polynomial convex body which has p-positive curvature function, then if , there exists an origin-symmetric convex body , such that
Proof Let be the generating function of . Since is p-polynomial, it follows from the proof of Theorem 3.5 that there exists an even function such that
The function must assume negative values, otherwise, by Theorem 1.1 we have , where is the convex body with . Let be a non-constant even function, such that: if , and if . By suitable approximation of the function F with spherical harmonics, we can find a nonnegative even function and an even function such that
Since K is a p-polynomial and has p-positive curvature, the surface area measure of K has a positive density . Thus, we can choose such that
By Theorem C, there exists an origin-symmetric convex body L such that
From (4.6) and Theorem 1.1, we see that .
Since , it follows that
Applying with (2.3), (4.5), (4.7), (2.10) and (4.6), we obtain
To complete the proof, we can use (2.6) to conclude
In particular, we replace by to Theorem 1.2, we have the following corollary, which was proved by Ryabogin and Zvavitch.
Corollary 4.5 
Let K and L be origin-symmetric convex bodies and where p is not an even integer. If L belongs to the class of projection bodies, then
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A project supported by Scientific Research Fund of Hunan Provincial Education Department (11C0542).
The author declares that they have no competing interests.