Open Access

L p Blaschke-Minkowski homomorphisms

Journal of Inequalities and Applications20132013:140

https://doi.org/10.1186/1029-242X-2013-140

Received: 25 March 2012

Accepted: 19 March 2013

Published: 2 April 2013

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.

Keywords

valuation L p Blaschke additionconvolution

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 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, 520].

An operator Z : K n K n is called a Minkowski valuation if
Z ( K L ) + Z ( K L ) = Z K + Z L ,
(1.1)

whenever K , L , K L K n , and here + is the Minkowski addition.

A Minkowski valuation Z is called SO ( n ) equivariant, if for all ϑ SO ( n ) and all K K n ,
Z ( ϑ K ) = ϑ Z K .
(1.2)
A Minkowski valuation Z is called homogeneity of degree p, if for all K K n and all λ 0 ,
Z ( λ K ) = λ p Z K .
(1.3)

A map Φ : K n K n is called a Blaschke-Minkowski homomorphism if it is continuous, SO ( n ) equivariant and satisfies Φ ( K # L ) = Φ K + Φ L , where # denotes the Blaschke addition, i.e., S ( K # L , ) = S ( K , ) + S ( L , ) .

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 Φ : K n K n be a Blaschke-Minkowski homomorphism, then there is a weakly positive g C ( S n 1 , e ˆ ) , unique up to a linear function, such that
h ( Φ K , ) = S ( K , ) g .

Theorem B [16]

Let Φ : K n K n be a Blaschke-Minkowski homomorphism. If K Φ K n and L K n , then
Φ K Φ L V ( K ) 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 [1013]; 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 G L ( n ) . Based on these results, in this article we study L p Blaschke-Minkowski homomorphisms which are continuous, ( n p 1 ) -homogeneous and SO ( n ) equivariant.

Theorem 1.1 Let p > 1 and p n . If Φ p : K e n K e n be an L p Blaschke-Minkowski homomorphism, then there is a nonnegative function g C ( S n 1 , e ˆ ) , such that
h p ( Φ p K , ) = S p ( K , ) g .
(1.4)
Theorem 1.2 Let 1 < p < n and p is not an even integer, and let Φ p : K e n K e n be an L p Blaschke-Minkowski homomorphism. If K K e n and L Φ p K e n , then
Φ p K Φ p L V ( K ) V ( L ) .
(1.5)
If p > n and p is not an even integer, then
Φ p K Φ p L V ( K ) V ( L ) ,
(1.6)

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

2 Notation and background material

Let K 0 n denote the set of convex bodies containing the origin in their interiors, and let K e n denote origin-symmetric convex bodies. In this paper, we restrict the dimension of R n to n 3 . A convex body K K n is uniquely determined by its support function, h ( K , ) . From the definition of h ( K , ) , it follows immediately that for λ > 0 and ϑ SO ( n ) ,
h ( λ K , u ) = λ h ( K , u ) and h ( ϑ K , u ) = h ( K , ϑ 1 u ) ,
(2.1)

where ϑ 1 is the inverse of ϑ.

For K , L K 0 n , p 1 , and ε > 0 , the L p Minkowski addition K + p ε L K 0 n is defined by (see [21])
h ( K + p ε L , ) p = h ( K , ) p + ε h ( L , ) p ,
(2.2)

where ‘  ’ in ε L denotes the Firey scalar multiplication, i.e., ε L = ε 1 p L .

If K , L K 0 n , then for p 1 , the L p mixed volume, V p ( K , L ) , of K and L is defined by (see [21])
V p ( K , L ) = lim ε 0 + V ( K + p ε L ) V ( K ) ε .
Corresponding to each K K 0 n , there is a positive Borel measure, S p ( K , ) , on S n 1 such that (see [21])
V p ( K , L ) = 1 n S n 1 h ( L , u ) p d S p ( K , u ) ,
(2.3)
for each L K 0 n . The measure S p ( K , ) is just the L p surface area measure of K, which is absolutely continuous with respect to classical surface area measure S ( K , ) , and has a Radon-Nikodym derivative
d S p ( K , ) d S ( K , ) = h ( K , ) 1 p .
(2.4)
A convex body K K 0 n is said to have a p-curvature function (see [21]) f p ( K , ) : S n 1 R , if its L p surface area measure S p ( K , ) is absolutely continuous with respect to spherical Lebesgue measure S and the Radon-Nikodym derivative
d S p ( K , ) d S = f p ( K , ) .
(2.5)
From the formula (2.3), it follows immediately that for each K 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 K 0 n , if p 1 , then
V p ( K , L ) V ( K ) n p n V ( L ) 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 n , there exists a unique origin-symmetric convex bodies K K e n , such that
S p ( K , ) = μ .
From (2.4), for λ > 0 , we have
S p ( λ K , ) = λ n p S p ( K , ) .
(2.7)
Noting the fact S ( ϑ K , ) = ϑ S ( K , ) for ϑ SO ( n ) and (2.1), one can obtain
S p ( ϑ K , ) = ϑ S p ( K , ) ,
(2.8)

where ϑ S p ( K , ) is the image measure of S p ( K , ) under the rotation ϑ. Obviously, S 1 ( K , ) is just S ( K , ) .

The L p Blaschke addition K # p L of K , L K 0 n is the convex body with
S p ( K # p L , ) = S p ( K , ) + S p ( L , ) .
(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 C ( SO ( n ) ) , C ( S n 1 ) be the spaces of continuous functions on SO ( n ) and S n 1 with uniform topology and M ( SO ( n ) ) , 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 C ( S n 1 ) and μ M ( S n 1 ) , we have ϑ f ( u ) = f ( ϑ 1 u ) , ϑ 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 e ˆ of S n 1 fixed. The projection from SO ( n ) onto S n 1 is ϑ ϑ ˆ : = ϑ e ˆ . Functions on S n 1 can be identified with right SO ( n 1 ) -invariant functions on SO ( n ) , by f ˇ ( ϑ ) = f ( ϑ ˆ ) , for f C ( S n 1 ) . In fact, C ( S n 1 ) is isomorphic to the subspace of right SO ( n 1 ) -invariant functions in C ( SO ( n ) ) .

The convolution μ f C ( S n 1 ) of a measure μ M ( SO ( n ) ) and a function f C ( S n 1 ) is defined by
( μ f ) ( u ) = SO ( n ) ϑ f ( u ) d μ ( ϑ ) .
(2.10)
The canonical pairing of f C ( S n 1 ) and μ M ( S n 1 ) is defined by
μ , f = f , μ = S n 1 f ( u ) d μ ( u ) .
(2.11)
A function f C ( S n 1 ) is called zonal, if ϑ f = f for every ϑ SO ( n 1 ) . Zonal functions depend only on the value u e ˆ . The set of continuous zonal functions on S n 1 will be denoted by C ( S n 1 , e ˆ ) and the definition of M ( S n 1 , e ˆ ) is analogous. A map Λ : C [ 1 , 1 ] C ( S n 1 , e ˆ ) is defined by
Λ f ( u ) = f ( u e ˆ ) , u 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 C ( S n 1 ) , μ M ( S n 1 , e ˆ ) and η SO ( n ) , then
( f μ ) ( η ˆ ) = S n 1 f ( η u ) d μ ( u ) .
(2.13)
If μ M ( S n 1 , e ˆ ) , for each f C ( S n 1 ) and every ϑ SO ( n ) , then
( ϑ f ) μ = ϑ ( f μ ) .
(2.14)

We denote 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 H k n . The space of all finite sums of spherical harmonics of dimension n is denoted by H n . The spaces H k n are pairwise orthogonal with respect to the usual inner product on C ( S n 1 ) . Clearly, H k n is invariant with respect to rotations.

Let P k n C [ 1 , 1 ] denote the Legendre polynomial of dimension n and order k. The zonal function Λ P k n is up to a multiplicative constant the unique zonal spherical harmonic in H k n . In each space H k n we choose an orthonormal basis H k 1 , , H k N ( n , k ) . The collection { H k 1 , , H k N ( n , k ) : k N } forms a complete orthogonal system in L 2 ( S n 1 ) . In particular, for every f L 2 ( S n 1 ) , the series
f k = 0 π k f
converges to f in the L 2 ( S n 1 ) -norm, where π k f H k n is the orthogonal projection of f on the space H k n . Using well-known properties of the Legendre polynomials, it is not hard to show that
π k f = N ( n , k ) ( f Λ P k n ) .
(2.15)
This leads to the spherical expansion of a measure μ M ( S n 1 ) ,
μ k = 0 π k μ ,
(2.16)
where π k μ H k n is defined by
π k μ = N ( n , k ) ( μ Λ 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 μ M ( S n 1 ) , the following special cases of (2.18):
π 0 μ = μ ( S n 1 ) and ( π 1 μ ) ( u ) = n S n 1 u v d μ ( v ) .
(2.18)
Let κ n denote the volume of the Euclidean unit ball B. By (2.3) and (2.19), for every convex body K K 0 n , it follows that
κ n π 0 h ( K , ) p = V p ( B , K ) and π 0 S p ( K , ) = n V p ( K , B ) .
(2.19)
A measure μ M ( S n 1 ) is uniquely determined by its series expansion (2.19). Using the fact that Λ P k n is (essentially) the unique zonal function in H k n , a simple calculation shows that for μ M ( S n 1 , e ˆ ) , formula (2.18) becomes
π k μ = N ( n , k ) μ , Λ P k n Λ P k n .
(2.20)

A zonal measure μ M ( S n 1 , e ˆ ) is defined by its so-called Legendre coefficients μ k : = μ , Λ P k n . Using π k H = H for every H H k n and the fact that spherical convolution of zonal measures is commutative, we have the Funk-Hecke theorem: If μ M ( S n 1 , e ˆ ) and H H k n , then H μ = μ k H .

A map Φ : D M ( S n 1 ) 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 N ,
π k Φ μ = c k π k μ , μ D .
(2.21)

From the Funk-Hecke theorem and the fact that the spherical convolution of zonal measures is commutative, it follows that, for μ M ( S n 1 , e ˆ ) , the map Φ μ : M ( S n 1 ) M ( S n 1 ) , defined by Φ μ = ν μ , 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 Ψ : K 0 n K 0 n is called an L p Minkowski valuation if
Ψ ( K L ) + p Ψ ( K L ) = Ψ K + p Ψ L ,
(3.1)

whenever K , L , K L K 0 n , and here ‘ + p ’ is L p Minkowski addition.

Definition 3.1 A map Φ p : K e n K e n satisfying the following properties (a), (b) and (c) is called an L p Blaschke-Minkowski homomorphism.
  1. (a)
    Φ p
    is continuous with respect to Hausdorff metric.
     
  2. (b)
    Φ p ( K # p L ) = Φ p K + p Φ p L
    for all K , L K e n .
     
  3. (c)
    Φ p
    is SO ( n ) equivariant, i.e., Φ p ( ϑ K ) = ϑ Φ p K for all ϑ SO ( n ) and all K 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 Φ : M ( S n 1 ) C ( S n 1 ) monotone, if non-negative measures are mapped to non-negative functions.

Lemma 3.1 A map Φ : M ( S n 1 ) C ( S n 1 ) is a monotone, linear map that is intertwines rotations if and only if there is a function f C ( S n 1 , e ˆ ) , such that
Φ μ = f μ .
(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 ϕ : M ( S n 1 ) R , μ Φ μ ( e ˆ ) . By the properties of Φ, the functional ϕ is positive and linear on M ( S n 1 ) , thus, by the Riesz representation theorem, there is a function f M + ( S n 1 ) such that
ϕ ( μ ) = S n 1 f ( u ) d μ ( u ) .
Since ϕ is SO ( n 1 ) invariant, the function f is zonal. Thus, we have for η SO ( n )
Φ μ ( η e ˆ ) = Φ ( η 1 μ ) ( e ˆ ) = ϕ ( η 1 μ ) = S n 1 f ( η u ) d μ ( u ) .

Lemma 3.1 follows now from (2.14). □

Proof of Theorem 1.1 Suppose that a map Φ p : K 0 n K 0 n satisfies h ( Φ p K , ) p = S p ( K , ) g , where g C ( S n 1 , e ˆ ) is a nonnegative measure. The continuity of Φ p follows from the fact that the support function h ( K , ) is continuous with respect to Hausdorff metric. From (2.9) and (2.1), for ϑ SO ( n ) , we obtain
h ( Φ p ϑ K , ) p = S p ( ϑ K , ) g = S p ( K , ϑ 1 ) g = h ( Φ p K , ϑ 1 ) p = h ( ϑ Φ p K , ) p .
Taking K = L in (1.4), we have
h ( Φ p L , ) p = S p ( L , ) g .
(3.3)
Combining with (2.2), (1.4) and (3.3), we obtain
h ( Φ p K + p Φ p L , ) p = h ( Φ p K , ) p + h ( Φ p L , ) p = S p ( K , ) g + S p ( L , ) g = ( S p ( K , ) + S p ( L , ) ) g = S p ( K # p L , ) g = h ( Φ p ( K # p L ) , ) p .
(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 Φ p , there is a function g C ( S n 1 , e ˆ ) 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 , ) S p ( L , ) , K , L K e n } coincides with M e ( S n 1 ) . The operator Φ ¯ : M ( S n 1 ) C ( S n 1 ) is defined by
Φ ¯ μ 1 = h ( Φ p K 1 , ) p h ( Φ p K 2 , ) p ,
(3.5)

where μ 1 = S p ( K 1 , ) S p ( K 2 , ) .

The operator Φ ¯ for μ 2 = S p ( L 1 , ) S p ( L 2 , ) immediately yields:
Φ ¯ μ 2 = h ( Φ p L 1 , ) p h ( Φ p L 2 , ) p .
(3.6)
Combining with (3.5), (3.6), (2.2) and (3.4), we obtain
Φ ¯ μ 1 + Φ ¯ μ 2 = h ( Φ p K 1 , ) p h ( Φ p K 2 , ) p + h ( Φ p L 1 , ) p h ( Φ p L 2 , ) p = h ( Φ p K 1 + p Φ p L 1 , ) p h ( Φ p K 2 + p Φ p L 2 , ) p = h ( Φ p ( K 1 # p L 1 ) , ) p h ( Φ p ( K 2 # p L 2 ) , ) p = Φ ¯ ( S p ( K 1 # p L 1 , ) S p ( K 2 # p L 2 , ) ) = Φ ¯ ( S p ( K 1 , ) + S p ( L 1 , ) S p ( K 2 , ) S p ( L 2 , ) ) = Φ ¯ ( μ 1 + μ 2 ) .

So, the operator Φ ¯ is linear.

Noting that Φ p is an L p Minkowski homomorphism and S p ( ϑ K , ) = ϑ S p ( K , ) , we obtain that the operator Φ ¯ is SO ( n ) equivariant.

Since the cone of the L p surface area measures of origin symmetric convex bodies is invariant under Φ ¯ , it is also monotone. Hence, by Lemma 3.1, there is a non-negative function g C ( S n 1 , e ˆ ) such that Φ ¯ μ = μ g . The statement now follows from
Φ ¯ S p ( K , ) = S p ( K , ) g = h ( Φ p K , ) p .

Hence, it is to complete the proof. □

Lutwak, Yang and Zhang first introduced the notion of L p -projection body (see [24]). Let Π p K , p 1 denote the compact convex symmetric set whose support function is given by
h ( Π p K , θ ) p = 1 n ω n c n 2 , p S p ( K , ) | θ , | p ,
(3.7)
where
c n , p = ω n + p ω 2 ω n ω p 1 .

Obviously, Π p : K e n K e n is an L p Blaschke-Minkowski homomorphism.

Lemma 3.2 [23]

If μ , ν M ( S n 1 ) and f C ( S n 1 ) , then
μ ν , f = μ , f ν .
Theorem 3.3 If Φ p : K e n K e n is an L p Blaschke-Minkowski homomorphism, then for K , L K e n ,
V p ( K , Φ p L ) = V p ( L , Φ p K ) .
(3.8)
Proof Let g C ( S n 1 , e ˆ ) be the generating function of Φ p . Using (2.3), Theorem 1.1 and Lemma 3.2, it follows that
n V p ( K , Φ p L ) = h ( Φ p L , ) p , S p ( K , ) = S p ( L , ) g , S p ( K , ) = S p ( L , ) , S p ( K , ) g = S p ( L , ) , h ( Φ p K , ) p = n V p ( L , Φ p K ) .
(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 Φ p is an L p Blaschke-Minkowski homomorphism, which is generated by the zonal function g, then for every origin symmetric convex body K K e n ,
π k h ( Φ p K , ) p = g k π k S p ( K , ) , k N ,
(3.10)

where the numbers g k are the Legendre coefficients of g, i.e., g k = g , Λ P k n .

Proof By (2.18) and Theorem 1.1, we have
π k h ( Φ p K , ) p = N ( n , k ) ( S p ( K , ) g Λ P k n ) .
Since spherical convolution is associative and g is zonal, we obtain from (2.18):
π k h ( Φ p K , ) p = g k N ( n , k ) ( S p ( K , ) Λ P k n ) = g k π k S p ( K , ) .

 □

Definition 3.2 If Φ p is an L p Blaschke-Minkowski homomorphism, generated by the zonal function g, then we call the subset K e n ( Φ p ) of K e n , defined by
K e n ( Φ p ) = { K K e n : π k S p ( K , ) = 0  if  g k = 0 } ,

the injectivity set of Φ 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 K e n p-polynomial if h ( K , ) p H n .

Clearly, the set of p-polynomial convex bodies is dense in K e n .

Let p > 1 and p n where p is not an even integer. The size of range, Φ p ( K e n ) , of the L p Blaschke-Minkowski homomorphism Φ p will be critical. The set of origin-symmetric convex bodies whose support functions are elements of the vector space
span { ( h ( Φ p K , ) p h ( Φ p L , ) p ) 1 p : K , L K e n }
(3.11)

is a large subset of K e n , provided the injectivity set K e n ( Φ p ) is not too small.

Theorem 3.5 Let p > 1 and p n where p is not an even integer. If Φ p : K e n K e n is an L p Blaschke-Minkowski homomorphism such that K e n K e n ( Φ p ) , then for every p-polynomial convex body K K e n , there exist origin-symmetry convex bodies K 1 , K 2 K e n such that
K + p Φ p K 1 = Φ p K 2 .
(3.12)
Proof Let K K e n be a p-polynomial convex body. From Definition 3.3, we have
h ( K , ) p = k = 0 m π k h ( K , ) p .
(3.13)
For K K e n and the properties of the orthogonal projection of f on the space H k n , we have π k h ( K , ) p = 0 for all odd k N . Let g C ( S n 1 , e ˆ ) denote the generating function of Φ and let g k denote the Legendre coefficients of g. From K e n K e n ( Φ ) and Definition 3.2, it follows that g k 0 for every even k N . We define
f : = k = 0 m c k π k h ( K , ) 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 g = k = 0 m c k g k π k h ( K , ) p = k = 0 m π k h ( K , ) p = h ( K , ) 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 , ) = f and S p ( K 2 , ) = f + . By Theorem 1.1 and (2.2), it follows that
K + p Φ p K 1 = Φ p K 2 .

 □

4 The Shephard-type problem

Let Φ p : K e n K e n denote a nontrivial L p Blaschke-Minkowski homomorphism, i.e., Φ p is continuous and SO ( n ) equivariant map satisfying Φ p ( K # p L ) = Φ p K + p Φ p L and Φ 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 n and Φ p : K 0 n K e n be an L p Blaschke-Minkowski homomorphism. Is there the implication:

If 0 < p < n , then
Φ p K Φ p L V ( K ) V ( L ) ?
(4.1)
If p > n , then
Φ p K Φ p L V ( K ) V ( L ) ?
(4.2)
Proof of Theorem 1.2 For L Φ p K e n and p is not an even integer, there exists an origin-symmetric convex body L 0 such that L = Φ 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 , Φ p L 0 ) = V p ( L 0 , Φ p K ) V p ( L 0 , Φ p L ) = V p ( L , Φ p L 0 ) = V ( L ) .
Applying the L p Minkowski inequality (2.6), we thus obtain that, if 1 < p < n , then
V ( K ) V ( L ) ,
and if p > n , then
V ( K ) 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 n , where p is not an even integer and Φ p : K e n K e n is an L p Blaschke-Minkowski homomorphism. If K , L Φ p K e n , then
Φ p K = Φ p L K = L .
(4.3)

Since the L p projection body operator Π 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 n , where p is not an even integer, and K and L are both L p projection bodies in R n . Then
Π p K = Π p L K = L .

Our next result shows that if the injectivity set K e n ( Φ p ) does not exhaust all of 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 K e n ( Φ p ) does not coincide with K e n , then there exist origin-symmetric convex bodies K , L K e n , such that
Φ p K Φ p L ,
but
V ( K ) > V ( L ) .

Proof Let g C ( S n 1 , e ˆ ) be the generating function of Φ p and let g k denote its Legendre coefficients. Since K e n ( Φ p ) K e n and Φ p is nontrivial, there exists, by Definition 3.2, an integer k N , such that g k = 0 and k 1 . We can choose α > 0 such that the function f ( u ) = 1 + α P k n ( u e ˆ ) , u S n 1 , is positive. According to Theorem C, there exists an origin-symmetric convex body L K e n with S p ( L , ) = f .

Since π k S p ( L , ) = π k ( 1 + α P k n ( u e ˆ ) ) 0 , from Definition 3.2 we have that L K e n ( Φ p ) .

From (2.20) and the properties of the orthogonal projection on the space H k n , we have that
n V p ( L , B ) = π 0 S p ( L , ) = 1 .
(4.4)
Using the fact that: For 1 < p < n where p is not an even integer, an origin-symmetric convex body L K e n ( Φ p ) is uniquely determined by its image Φ p L , we obtain that Φ p L = Φ 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 = 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 K e n K e n ( Φ p ) . If K K e n is a p-polynomial convex body which has p-positive curvature function, then if K Φ p K e n , there exists an origin-symmetric convex body L K e n , such that
Φ p K Φ p L ,
but
V ( K ) > V ( L ) .
Proof Let g C ( S n 1 , e ˆ ) be the generating function of Φ p . Since K K e n is p-polynomial, it follows from the proof of Theorem 3.5 that there exists an even function f H n such that
h ( K , ) p = f g .
(4.5)
The function must assume negative values, otherwise, by Theorem 1.1 we have K = Φ p K 0 , where K 0 is the convex body with S p ( K 0 , ) = f . Let F C ( S n 1 ) be a non-constant even function, such that: F ( u ) 0 if f ( u ) < 0 , and F ( u ) = 0 if f ( u ) 0 . By suitable approximation of the function F with spherical harmonics, we can find a nonnegative even function G H n and an even function H H n such that
f , G < 0 , and G = H 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 , ) . Thus, we can choose α > 0 such that
S p ( K , ) + α H > 0 .
By Theorem C, there exists an origin-symmetric convex body L such that
S p ( L , ) = S p ( K , ) + α H .
(4.7)

From (4.6) and Theorem 1.1, we see that h ( Φ p L , ) p = h ( Φ p K , ) p + α G .

Since G 0 , it follows that
Φ p K Φ p L .
(4.8)
Applying with (2.3), (4.5), (4.7), (2.10) and (4.6), we obtain
n ( V p ( K , L ) V ( K ) ) = h ( K , ) p , S p ( L , ) S p ( K , ) = h ( K , ) p , α H = α f g , H = α f , H g = α f , G < 0 .
(4.9)
To complete the proof, we can use (2.6) to conclude
V ( K ) > V ( L ) .

 □

In particular, we replace Φ p by Π 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 p < n where p is not an even integer. If L belongs to the class of L p projection bodies, then
Π p K Π p L V ( K ) V ( L ) .

Declarations

Acknowledgements

A project supported by Scientific Research Fund of Hunan Provincial Education Department (11C0542).

Authors’ Affiliations

(1)
School of Mathematics and Computational Science, Hunan University of Science and Technology

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