Skip to main content

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 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(KL)+Z(KL)=ZK+ZL,
(1.1)

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

A Minkowski valuation Z is called SO(n) equivariant, if for all ϑSO(n) and all K K n ,

Z(ϑK)=ϑZK.
(1.2)

A Minkowski valuation Z is called homogeneity of degree p, if for all K K n and all λ0,

Z(λK)= λ p ZK.
(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)=ΦK+ΦL, 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 (n1)-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 gC( S n 1 , e ˆ ), unique up to a linear function, such that

h(ΦK,)=S(K,)g.

Theorem B [16]

Let Φ: K n K n be a Blaschke-Minkowski homomorphism. If KΦ K n and L K n , then

ΦKΦLV(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 GL(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 pn. If Φ p : K e n K e n be an L p Blaschke-Minkowski homomorphism, then there is a nonnegative function gC( 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 LV(K)V(L).
(1.5)

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

Φ p K Φ p LV(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 n3. 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 ϑSO(n),

h(λK,u)=λh(K,u)andh(ϑK,u)=h ( K , ϑ 1 u ) ,
(2.1)

where ϑ 1 is the inverse of ϑ.

For K,L K 0 n , p1, and ε>0, the L p Minkowski addition K + p εL K 0 n is defined by (see [21])

h ( K + p ε L , ) p =h ( K , ) p +εh ( L , ) p ,
(2.2)

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

If K,L K 0 n , then for p1, 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 p1, 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 pn, 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 (λK,)= λ n p S p (K,).
(2.7)

Noting the fact S(ϑK,)=ϑS(K,) for ϑSO(n) and (2.1), one can obtain

S p (ϑK,)=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 fC( S n 1 ) and μM( S n 1 ), we have ϑf(u)=f( ϑ 1 u), ϑSO(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(n1), where SO(n1) 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(n1)-invariant functions on SO(n), by f ˇ (ϑ)=f( ϑ ˆ ), for fC( S n 1 ). In fact, C( S n 1 ) is isomorphic to the subspace of right SO(n1)-invariant functions in C(SO(n)).

The convolution μfC( S n 1 ) of a measure μM(SO(n)) and a function fC( S n 1 ) is defined by

(μf)(u)= SO ( n ) ϑf(u)dμ(ϑ).
(2.10)

The canonical pairing of fC( S n 1 ) and μM( S n 1 ) is defined by

μ,f=f,μ= S n 1 f(u)dμ(u).
(2.11)

A function fC( S n 1 ) is called zonal, if ϑf=f for every ϑSO(n1). 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

Λf(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 fC( S n 1 ), μM( S n 1 , e ˆ ) and ηSO(n), then

(fμ)( η ˆ )= S n 1 f(ηu)dμ(u).
(2.13)

If μM( S n 1 , e ˆ ), for each fC( S n 1 ) and every ϑSO(n), then

(ϑf)μ=ϑ(fμ).
(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 ) :kN} 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 μM( 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 μM( 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 uvdμ(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 μM( 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 μM( 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 μM( 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 μM( S n 1 , e ˆ ) and H H k n , then Hμ= μ k H.

A map Φ:DM( 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 kN,

π k Φμ= c k π k μ,μD.
(2.21)

From the Funk-Hecke theorem and the fact that the spherical convolution of zonal measures is commutative, it follows that, for μM( S n 1 , e ˆ ), the map Φ μ :M( S n 1 )M( S n 1 ), defined by Φ μ =νμ, 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(KL) + p Ψ(KL)=ΨK + p ΨL,
(3.1)

whenever K,L,KL 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 (ϑK)=ϑ Φ p K for all ϑSO(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 fC( S n 1 , e ˆ ), such that

Φμ=fμ.
(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, μΦμ( 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)dμ(u).

Since ϕ is SO(n1) invariant, the function f is zonal. Thus, we have for ηSO(n)

Φμ(η e ˆ )=Φ ( η 1 μ ) ( e ˆ )=ϕ ( η 1 μ ) = S n 1 f(ηu)dμ(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 gC( 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 ϑSO(n), we obtain

h ( Φ p ϑ K , ) p = S p (ϑK,)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 gC( 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 (ϑK,)=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 gC( S n 1 , e ˆ ) such that Φ ¯ μ=μg. 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, p1 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,νM( S n 1 ) and fC( S n 1 ), then

μν,f=μ,fν.

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 gC( 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,),kN,
(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 pn 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 pn 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 kN. Let gC( 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 kN. 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

fg= 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, pn 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 LV(K)V(L)?
(4.1)

If p>n, then

Φ p K Φ p LV(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, pn, 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 LK=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, pn, 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 LK=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 gC( 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 kN, such that g k =0 and k1. 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 gC( 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 =fg.
(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 FC( 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,andG=Hg.
(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,)+αH>0.

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

S p (L,)= S p (K,)+αH.
(4.7)

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

Since G0, 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 1p<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 LV(K)V(L).

References

  1. Hadwiger H: Vorlesungenuber Inhalt, Oberflache und Isoperimetrie. Springer, Berlin; 1957.

    Book  Google Scholar 

  2. Schneider R: Equivariant endomorphisms of the space of convex bodies. Trans. Am. Math. Soc. 1974, 194: 53–78.

    Article  Google Scholar 

  3. McMullen P, Schneider R: Valuations on convex bodies. In Convexity and Its Applications. Edited by: Gruber PM, Wills JM. Birkhäuser, Basel; 1983:170–247.

    Chapter  Google Scholar 

  4. Schneider R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge; 1993.

    Book  Google Scholar 

  5. Alesker S: Continuous rotation invariant valuations on convex sets. Ann. Math. 1999, 149: 977–1005. 10.2307/121078

    MathSciNet  Article  Google Scholar 

  6. Alesker S: Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 2001, 11: 244–272. 10.1007/PL00001675

    MathSciNet  Article  Google Scholar 

  7. Haberl C: Star body valued valuations. Indiana Univ. Math. J. 2009, 58: 2253–2276. 10.1512/iumj.2009.58.3685

    MathSciNet  Article  Google Scholar 

  8. Haberl C, Ludwig M:A characterization of L p intersection bodies. Int. Math. Res. Not. 2006., 2006: Article ID 10548

    Google Scholar 

  9. Klain DA: Star valuations and dual mixed volumes. Adv. Math. 1996, 121: 80–101. 10.1006/aima.1996.0048

    MathSciNet  Article  Google Scholar 

  10. Ludwig M: Projection bodies and valuations. Adv. Math. 2002, 172(2):158–168. 10.1016/S0001-8708(02)00021-X

    MathSciNet  Article  Google Scholar 

  11. Ludwig M: Minkowski valuations. Trans. Am. Math. Soc. 2005, 357(10):4191–4213. 10.1090/S0002-9947-04-03666-9

    MathSciNet  Article  Google Scholar 

  12. Ludwig M: Intersection bodies and valuations. Am. J. Math. 2006, 128(6):1409–1428. 10.1353/ajm.2006.0046

    MathSciNet  Article  Google Scholar 

  13. Ludwig M, Reitzner M:A classification of SL(n) invariant valuations. Ann. Math. 2010, 172: 1223–1271. 10.4007/annals.2010.172.1223

    MathSciNet  Article  Google Scholar 

  14. Schneider R, Schuster FE: Rotation equivariant Minkowski valuations. Int. Math. Res. Not. 2006., 2006: Article ID 72894

    Google Scholar 

  15. Schuster FE: Convolutions and multiplier transformations of convex bodies. Trans. Am. Math. Soc. 2007, 359(11):5567–5591. 10.1090/S0002-9947-07-04270-5

    Article  Google Scholar 

  16. Schuster FE: Valuations and Busemann-Petty type problems. Adv. Math. 2008, 219(1):344–368. 10.1016/j.aim.2008.05.001

    MathSciNet  Article  Google Scholar 

  17. Schuster FE: Crofton measures and Minkowski valuations. Duke Math. J. 2010, 154: 1–30. 10.1215/00127094-2010-033

    MathSciNet  Article  Google Scholar 

  18. Schuster FE, Wannerer T:GL(n) contravariant Minkowski valuations. Trans. Am. Math. Soc. 2012, 364: 815–826. 10.1090/S0002-9947-2011-05364-X

    MathSciNet  Article  Google Scholar 

  19. Wang W, Liu LJ, He BW: L p radial Minkowski homomorphisms. Taiwan. J. Math. 2011, 15(3):1183–1199.

    MathSciNet  Google Scholar 

  20. Wannerer T:GL(n) equivariant Minkowski valuations. Indiana Univ. Math. J. 2011, 60: 1655–1672. 10.1512/iumj.2011.60.4425

    MathSciNet  Article  Google Scholar 

  21. Lutwak E: The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas. Adv. Math. 1996, 118: 244–294. 10.1006/aima.1996.0022

    MathSciNet  Article  Google Scholar 

  22. Lutwak E: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 1993, 38: 131–150.

    MathSciNet  Google Scholar 

  23. Grinberg E, Zhang G: Convolutions, transforms and convex bodies. Proc. Lond. Math. Soc. 1999, 78: 77–115. 10.1112/S0024611599001653

    MathSciNet  Article  Google Scholar 

  24. Lutwak E, Yang D, Zhang G: L p affine isoperimetric inequalities. J. Differ. Geom. 2000, 56: 111–132.

    MathSciNet  Google Scholar 

  25. Ryabogin D, Zvavitch A: The Fourier transform and Firey projections of convex bodies. Indiana Univ. Math. J. 2004, 53: 234–241.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei Wang.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Wang, W. L p Blaschke-Minkowski homomorphisms. J Inequal Appl 2013, 140 (2013). https://doi.org/10.1186/1029-242X-2013-140

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

  • valuation
  • L p Blaschke addition
  • convolution