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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2013 | Research

$L_{p}$ Blaschke-Minkowski homomorphisms

verfasst von: Wei Wang

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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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.

Competing interests

The author declares that they have no competing interests.

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, 5‐20].

An operator

Z : K^{n} \to K^{n}

is called a Minkowski valuation if

Z (K \cup L) + Z (K \cap L) = Z K + Z L,

(1.1)

whenever

K, L, K \cup L \in K^{n}

, and here + is the Minkowski addition.

A Minkowski valuation Z is called

SO (n)

equivariant, if for all

ϑ \in SO (n)

and all

K \in K^{n}

Z (ϑ K) = ϑ Z K .

(1.2)

A Minkowski valuation Z is called homogeneity of degree p, if for all

K \in K^{n}

and all

λ \geq 0

Z (λ K) = λ^{p} Z K .

(1.3)

A map

Φ : K^{n} \to 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, \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]

Φ : K^{n} \to K^{n}

be a Blaschke-Minkowski homomorphism, then there is a weakly positive

g \in C (S^{n - 1}, \hat{e})

, unique up to a linear function, such that

h (Φ K, \cdot) = S (K, \cdot) * g .

Theorem B [16]

Let

Φ : K^{n} \to K^{n}

be a Blaschke-Minkowski homomorphism. If

K \in Φ K^{n}

and

L \in K^{n}

, then

Φ K \subseteq Φ L \Rightarrow V (K) \leq 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

G L (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 \neq n

. If

Φ_{p} : K_{e}^{n} \to K_{e}^{n}

be an

L_{p}

Blaschke-Minkowski homomorphism, then there is a nonnegative function

g \in C (S^{n - 1}, \hat{e})

, such that

h^{p} (Φ_{p} K, \cdot) = S_{p} (K, \cdot) * g .

(1.4)

Theorem 1.2 Let

1 < p < n

and p is not an even integer, and let

Φ_{p} : K_{e}^{n} \to K_{e}^{n}

be an

L_{p}

Blaschke-Minkowski homomorphism. If

K \in K_{e}^{n}

and

L \in Φ_{p} K_{e}^{n}

, then

Φ_{p} K \subseteq Φ_{p} L \Rightarrow V (K) \leq V (L) .

(1.5)

p > n

and p is not an even integer, then

Φ_{p} K \subseteq Φ_{p} L \Rightarrow V (K) \geq 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}

n \geq 3

. A convex body

K \in K^{n}

is uniquely determined by its support function,

h (K, \cdot)

. From the definition of

h (K, \cdot)

, it follows immediately that for

λ > 0

and

ϑ \in 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 \in K_{0}^{n}

p \geq 1

, and

ε > 0

, the

L_{p}

Minkowski addition

K +_{p} ε \cdot L \in K_{0}^{n}

is defined by (see [21])

h {(K +_{p} ε \cdot L, \cdot)}^{p} = h {(K, \cdot)}^{p} + ε h {(L, \cdot)}^{p},

(2.2)

where ‘ ⋅ ’ in

ε \cdot L

denotes the Firey scalar multiplication, i.e.,

ε \cdot L = ε^{\frac{1}{p}} L

K, L \in K_{0}^{n}

, then for

p \geq 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 \to 0^{+}} \frac{V (K +_{p} ε \cdot L) - V (K)}{ε} .

Corresponding to each

K \in 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 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)}{d S (K, \cdot)} = h {(K, \cdot)}^{1 - p} .

(2.4)

A convex body

K \in K_{0}^{n}

is said to have a p-curvature function (see [21])

f_{p} (K, \cdot) : S^{n - 1} \to 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)}{d S} = 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 K_{0}^{n}

, if

p \geq 1

, then

V_{p} (K, L) \geq V {(K)}^{\frac{n - p}{n}} V {(L)}^{\frac{p}{n}},

(2.6)

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 \neq n

, there exists a unique origin-symmetric convex bodies

K \in K_{e}^{n}

, such that

S_{p} (K, \cdot) = μ .

From (2.4), for

λ > 0

, we have

S_{p} (λ K, \cdot) = λ^{n - p} S_{p} (K, \cdot) .

(2.7)

Noting the fact

S (ϑ K, \cdot) = ϑ S (K, \cdot)

for

ϑ \in SO (n)

and (2.1), one can obtain

S_{p} (ϑ K, \cdot) = ϑ S_{p} (K, \cdot),

(2.8)

where

ϑ 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 #_{p} L

K, L \in K_{0}^{n}

is the convex body with

S_{p} (K #_{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

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 \in C (S^{n - 1})

and

μ \in M (S^{n - 1})

, we have

ϑ f (u) = f (ϑ^{- 1} u)

ϑ \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}

S^{n - 1}

fixed. The projection from

SO (n)

onto

S^{n - 1}

ϑ \mapsto \hat{ϑ} : = ϑ \hat{e}

. Functions on

S^{n - 1}

can be identified with right

SO (n - 1)

-invariant functions on

SO (n)

, by

\overset{ˇ}{f} (ϑ) = f (\hat{ϑ})

, for

f \in 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 \in C (S^{n - 1})

of a measure

μ \in M (SO (n))

and a function

f \in C (S^{n - 1})

is defined by

(μ * f) (u) = \int_{SO (n)} ϑ f (u) d μ (ϑ) .

(2.10)

The canonical pairing of

f \in C (S^{n - 1})

and

μ \in M (S^{n - 1})

is defined by

〈 μ, f 〉 = 〈 f, μ 〉 = \int_{S^{n - 1}} f (u) d μ (u) .

(2.11)

A function

f \in C (S^{n - 1})

is called zonal, if

ϑ f = f

for every

ϑ \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

C (S^{n - 1}, \hat{e})

and the definition of

M (S^{n - 1}, \hat{e})

is analogous. A map

Λ : C [- 1, 1] \to C (S^{n - 1}, \hat{e})

is defined by

Λ 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 C (S^{n - 1})

μ \in M (S^{n - 1}, \hat{e})

and

η \in SO (n)

, then

(f * μ) (\hat{η}) = \int_{S^{n - 1}} f (η u) d μ (u) .

(2.13)

μ \in M (S^{n - 1}, \hat{e})

, for each

f \in C (S^{n - 1})

and every

ϑ \in 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} \in 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}, \dots, H_{k N (n, k)}

. The collection

{H_{k 1}, \dots, H_{k N (n, k)} : k \in N}

forms a complete orthogonal system in

L^{2} (S^{n - 1})

. In particular, for every

f \in L^{2} (S^{n - 1})

, the series

f \sim \sum_{k = 0}^{\infty} π_{k} f

converges to f in the

L^{2} (S^{n - 1})

-norm, where

π_{k} f \in 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 \in M (S^{n - 1})

μ \sim \sum_{k = 0}^{\infty} π_{k} μ,

(2.16)

where

π_{k} μ \in 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 \in M (S^{n - 1})

, the following special cases of (2.18):

π_{0} μ = μ (S^{n - 1}) and (π_{1} μ) (u) = n \int_{S^{n - 1}} u \cdot 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 \in K_{0}^{n}

, it follows that

κ_{n} π_{0} h {(K, \cdot)}^{p} = V_{p} (B, K) and π_{0} S_{p} (K, \cdot) = n V_{p} (K, B) .

(2.19)

A measure

μ \in 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 \in M (S^{n - 1}, \hat{e})

, 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 \in M (S^{n - 1}, \hat{e})

is defined by its so-called Legendre coefficients

μ_{k} : = 〈 μ, Λ P_{k}^{n} 〉

. Using

π_{k} H = H

for every

H \in H_{k}^{n}

and the fact that spherical convolution of zonal measures is commutative, we have the Funk-Hecke theorem: If

μ \in M (S^{n - 1}, \hat{e})

and

H \in H_{k}^{n}

, then

H * μ = μ_{k} H

A map

Φ : D \subseteq M (S^{n - 1}) \to 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 N

π_{k} Φ μ = c_{k} π_{k} μ, \forall μ \in 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 \in M (S^{n - 1}, \hat{e})

, the map

Φ_{μ} : M (S^{n - 1}) \to 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} \to K_{0}^{n}

is called an

L_{p}

Minkowski valuation if

Ψ (K \cup L) +_{p} Ψ (K \cap L) = Ψ K +_{p} Ψ L,

(3.1)

whenever

K, L, K \cup L \in K_{0}^{n}

, and here ‘

+_{p}

’ is

L_{p}

Minkowski addition.

Definition 3.1 A map

Φ_{p} : K_{e}^{n} \to K_{e}^{n}

satisfying the following properties (a), (b) and (c) is called an

L_{p}

Blaschke-Minkowski homomorphism.

(a)

Φ_{p}

is continuous with respect to Hausdorff metric.

(b)

Φ_{p} (K #_{p} L) = Φ_{p} K +_{p} Φ_{p} L

for all

K, L \in K_{e}^{n}

(c)

Φ_{p}

SO (n)

equivariant, i.e.,

Φ_{p} (ϑ K) = ϑ Φ_{p} K

for all

ϑ \in SO (n)

and all

K \in 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}) \to 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}) \to C (S^{n - 1})

is a monotone, linear map that is intertwines rotations if and only if there is a function

f \in C (S^{n - 1}, \hat{e})

, 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}) \to R

μ \to Φ μ (\hat{e})

. 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 \in M_{+} (S^{n - 1})

such that

ϕ (μ) = \int_{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 \in SO (n)

Φ μ (η \hat{e}) = Φ (η^{- 1} μ) (\hat{e}) = ϕ (η^{- 1} μ) = \int_{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} \to K_{0}^{n}

satisfies

h {(Φ_{p} K, \cdot)}^{p} = S_{p} (K, \cdot) * g

, where

g \in C (S^{n - 1}, \hat{e})

is a nonnegative measure. The continuity of

Φ_{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

ϑ \in SO (n)

, we obtain

h {(Φ_{p} ϑ K, \cdot)}^{p} = S_{p} (ϑ K, \cdot) * g = S_{p} (K, ϑ^{- 1} \cdot) * g = h {(Φ_{p} K, ϑ^{- 1} \cdot)}^{p} = h {(ϑ Φ_{p} K, \cdot)}^{p} .

Taking

K = L

in (1.4), we have

h {(Φ_{p} L, \cdot)}^{p} = S_{p} (L, \cdot) * g .

(3.3)

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

\begin{aligned} h {(Φ_{p} K +_{p} Φ_{p} L, \cdot)}^{p} & = h {(Φ_{p} K, \cdot)}^{p} + h {(Φ_{p} L, \cdot)}^{p} \\ = S_{p} (K, \cdot) * g + S_{p} (L, \cdot) * g \\ = (S_{p} (K, \cdot) + S_{p} (L, \cdot)) * g \\ = S_{p} (K #_{p} L, \cdot) * g \\ = h {(Φ_{p} (K #_{p} L), \cdot)}^{p} . \end{aligned}

(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 \in 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 K_{e}^{n}}

coincides with

M_{e} (S^{n - 1})

. The operator

\bar{Φ} : M (S^{n - 1}) \to C (S^{n - 1})

is defined by

\bar{Φ} μ_{1} = h {(Φ_{p} K_{1}, \cdot)}^{p} - h {(Φ_{p} K_{2}, \cdot)}^{p},

(3.5)

where

μ_{1} = S_{p} (K_{1}, \cdot) - S_{p} (K_{2}, \cdot)

The operator

\bar{Φ}

for

μ_{2} = S_{p} (L_{1}, \cdot) - S_{p} (L_{2}, \cdot)

immediately yields:

\bar{Φ} μ_{2} = h {(Φ_{p} L_{1}, \cdot)}^{p} - h {(Φ_{p} L_{2}, \cdot)}^{p} .

(3.6)

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

\begin{aligned} \bar{Φ} μ_{1} + \bar{Φ} μ_{2} & = h {(Φ_{p} K_{1}, \cdot)}^{p} - h {(Φ_{p} K_{2}, \cdot)}^{p} + h {(Φ_{p} L_{1}, \cdot)}^{p} - h {(Φ_{p} L_{2}, \cdot)}^{p} \\ = h {(Φ_{p} K_{1} +_{p} Φ_{p} L_{1}, \cdot)}^{p} - h {(Φ_{p} K_{2} +_{p} Φ_{p} L_{2}, \cdot)}^{p} \\ = h {(Φ_{p} (K_{1} #_{p} L_{1}), \cdot)}^{p} - h {(Φ_{p} (K_{2} #_{p} L_{2}), \cdot)}^{p} \\ = \bar{Φ} (S_{p} (K_{1} #_{p} L_{1}, \cdot) - S_{p} (K_{2} #_{p} L_{2}, \cdot)) \\ = \bar{Φ} (S_{p} (K_{1}, \cdot) + S_{p} (L_{1}, \cdot) - S_{p} (K_{2}, \cdot) - S_{p} (L_{2}, \cdot)) \\ = \bar{Φ} (μ_{1} + μ_{2}) . \end{aligned}

So, the operator

\bar{Φ}

is linear.

Noting that

Φ_{p}

is an

L_{p}

Minkowski homomorphism and

S_{p} (ϑ K, \cdot) = ϑ S_{p} (K, \cdot)

, we obtain that the operator

\bar{Φ}

SO (n)

equivariant.

Since the cone of the

L_{p}

surface area measures of origin symmetric convex bodies is invariant under

\bar{Φ}

, it is also monotone. Hence, by Lemma 3.1, there is a non-negative function

g \in C (S^{n - 1}, \hat{e})

such that

\bar{Φ} μ = μ * g

. The statement now follows from

\bar{Φ} S_{p} (K, \cdot) = S_{p} (K, \cdot) * g = h {(Φ_{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

Π_{p} K

p \geq 1

denote the compact convex symmetric set whose support function is given by

h {(Π_{p} K, θ)}^{p} = \frac{1}{n ω_{n} c_{n - 2, p}} S_{p} (K, \cdot) * {| 〈 θ, \cdot 〉 |}^{p},

(3.7)

where

c_{n, p} = \frac{ω_{n + p}}{ω_{2} ω_{n} ω_{p - 1}} .

Obviously,

Π_{p} : K_{e}^{n} \to K_{e}^{n}

is an

L_{p}

Blaschke-Minkowski homomorphism.

Lemma 3.2 [23]

μ, ν \in M (S^{n - 1})

and

f \in C (S^{n - 1})

, then

〈 μ * ν, f 〉 = 〈 μ, f * ν 〉 .

Theorem 3.3 If

Φ_{p} : K_{e}^{n} \to K_{e}^{n}

is an

L_{p}

Blaschke-Minkowski homomorphism, then for

K, L \in K_{e}^{n}

V_{p} (K, Φ_{p} L) = V_{p} (L, Φ_{p} K) .

(3.8)

Proof Let

g \in C (S^{n - 1}, \hat{e})

be the generating function of

Φ_{p}

. Using (2.3), Theorem 1.1 and Lemma 3.2, it follows that

\begin{aligned} n V_{p} (K, Φ_{p} L) & = 〈 h {(Φ_{p} L, \cdot)}^{p}, S_{p} (K, \cdot) 〉 \\ = 〈 S_{p} (L, \cdot) * g, S_{p} (K, \cdot) 〉 \\ = 〈 S_{p} (L, \cdot), S_{p} (K, \cdot) * g 〉 \\ = 〈 S_{p} (L, \cdot), h {(Φ_{p} K, \cdot)}^{p} 〉 \\ = n V_{p} (L, Φ_{p} K) . \end{aligned}

(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 \in K_{e}^{n}

π_{k} h {(Φ_{p} K, \cdot)}^{p} = g_{k} π_{k} S_{p} (K, \cdot), k \in 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, \cdot)}^{p} = N (n, k) (S_{p} (K, \cdot) * 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, \cdot)}^{p} = g_{k} N (n, k) (S_{p} (K, \cdot) * Λ P_{k}^{n}) = g_{k} π_{k} S_{p} (K, \cdot) .

□

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})

K_{e}^{n}

, defined by

K_{e}^{n} (Φ_{p}) = {K \in K_{e}^{n} : π_{k} S_{p} (K, \cdot) = 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 \in K_{e}^{n}

p-polynomial if

h {(K, \cdot)}^{p} \in H^{n}

Clearly, the set of p-polynomial convex bodies is dense in

K_{e}^{n}

Let

p > 1

and

p \neq 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, \cdot)}^{p} - h {(Φ_{p} L, \cdot)}^{p})}^{\frac{1}{p}} : K, L \in 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 \neq n

where p is not an even integer. If

Φ_{p} : K_{e}^{n} \to K_{e}^{n}

is an

L_{p}

Blaschke-Minkowski homomorphism such that

K_{e}^{n} \subseteq K_{e}^{n} (Φ_{p})

, then for every p-polynomial convex body

K \in K_{e}^{n}

, there exist origin-symmetry convex bodies

K_{1}, K_{2} \in K_{e}^{n}

such that

K +_{p} Φ_{p} K_{1} = Φ_{p} K_{2} .

(3.12)

Proof Let

K \in K_{e}^{n}

be a p-polynomial convex body. From Definition 3.3, we have

h {(K, \cdot)}^{p} = \sum_{k = 0}^{m} π_{k} h {(K, \cdot)}^{p} .

(3.13)

For

K \in K_{e}^{n}

and the properties of the orthogonal projection of f on the space

H_{k}^{n}

, we have

π_{k} h {(K, \cdot)}^{p} = 0

for all odd

k \in N

. Let

g \in C (S^{n - 1}, \hat{e})

denote the generating function of Φ and let

g_{k}

denote the Legendre coefficients of g. From

K_{e}^{n} \subseteq K_{e}^{n} (Φ)

and Definition 3.2, it follows that

g_{k} \neq 0

for every even

k \in N

. We define

f : = \sum_{k = 0}^{m} c_{k} π_{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 * g = \sum_{k = 0}^{m} c_{k} g_{k} π_{k} h {(K, \cdot)}^{p} = \sum_{k = 0}^{m} π_{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} Φ_{p} K_{1} = Φ_{p} K_{2} .

□

4 The Shephard-type problem

Let

Φ_{p} : K_{e}^{n} \to 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 \neq n

and

Φ_{p} : K_{0}^{n} \to K_{e}^{n}

be an

L_{p}

Blaschke-Minkowski homomorphism. Is there the implication:

0 < p < n

, then

Φ_{p} K \subseteq Φ_{p} L \Rightarrow V (K) \leq V (L) ?

(4.1)

p > n

, then

Φ_{p} K \subseteq Φ_{p} L \Rightarrow V (K) \geq V (L) ?

(4.2)

Proof of Theorem 1.2 For

L \in 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) \leq 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) \leq V (L),

and if

p > n

, then

V (K) \geq 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 \neq n

, where p is not an even integer and

Φ_{p} : K_{e}^{n} \to K_{e}^{n}

is an

L_{p}

Blaschke-Minkowski homomorphism. If

K, L \in Φ_{p} K_{e}^{n}

, then

Φ_{p} K = Φ_{p} L \Leftrightarrow 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 \neq 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 \Leftrightarrow 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 \in K_{e}^{n}

, such that

Φ_{p} K \subseteq Φ_{p} L,

but

V (K) > V (L) .

Proof Let

g \in C (S^{n - 1}, \hat{e})

be the generating function of

Φ_{p}

and let

g_{k}

denote its Legendre coefficients. Since

K_{e}^{n} (Φ_{p}) \neq K_{e}^{n}

and

Φ_{p}

is nontrivial, there exists, by Definition 3.2, an integer

k \in N

, such that

g_{k} = 0

and

k \geq 1

. We can choose

α > 0

such that the function

f (u) = 1 + α 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 K_{e}^{n}

with

S_{p} (L, \cdot) = f

Since

π_{k} S_{p} (L, \cdot) = π_{k} (1 + α P_{k}^{n} (u \cdot \hat{e})) \neq 0

, from Definition 3.2 we have that

L \notin 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, \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 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} = \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

K_{e}^{n} \subseteq K_{e}^{n} (Φ_{p})

. If

K \in K_{e}^{n}

is a p-polynomial convex body which has p-positive curvature function, then if

K \notin Φ_{p} K_{e}^{n}

, there exists an origin-symmetric convex body

L \in K_{e}^{n}

, such that

Φ_{p} K \subseteq Φ_{p} L,

but

V (K) > V (L) .

Proof Let

g \in C (S^{n - 1}, \hat{e})

be the generating function of

Φ_{p}

. Since

K \in K_{e}^{n}

is p-polynomial, it follows from the proof of Theorem 3.5 that there exists an even function

f \in H^{n}

such that

h {(K, \cdot)}^{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}, \cdot) = f

. Let

F \in C (S^{n - 1})

be a non-constant even function, such that:

F (u) \geq 0

f (u) < 0

, and

F (u) = 0

f (u) \geq 0

. By suitable approximation of the function F with spherical harmonics, we can find a nonnegative even function

G \in H^{n}

and an even function

H \in 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, \cdot)

. Thus, we can choose

α > 0

such that

S_{p} (K, \cdot) + α H > 0 .

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

S_{p} (L, \cdot) = S_{p} (K, \cdot) + α H .

(4.7)

From (4.6) and Theorem 1.1, we see that

h {(Φ_{p} L, \cdot)}^{p} = h {(Φ_{p} K, \cdot)}^{p} + α G

Since

G \geq 0

, it follows that

Φ_{p} K \subseteq Φ_{p} L .

(4.8)

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

\begin{aligned} n (V_{p} (K, L) - V (K)) & = 〈 h {(K, \cdot)}^{p}, S_{p} (L, \cdot) - S_{p} (K, \cdot) 〉 \\ = 〈 h {(K, \cdot)}^{p}, α H 〉 \\ = α 〈 f * g, H 〉 \\ = α 〈 f, H * g 〉 \\ = α 〈 f, G 〉 < 0 . \end{aligned}

(4.9)

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

V (K) > V (L) .

□

In particular, we replace

Φ_{p}

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 \leq p < n

where p is not an even integer. If L belongs to the class of

L_{p}

projection bodies, then

Π_{p} K \subseteq Π_{p} L \Rightarrow V (K) \leq V (L) .

Acknowledgements

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

Open AccessThis 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.

Competing interests

The author declares that they have no competing interests.

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Titel: Blaschke-Minkowski homomorphisms
verfasst von: Wei Wang
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-140

Springer Professional

$L_{p}$ Blaschke-Minkowski homomorphisms

Abstract

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1 Introduction

2 Notation and background material

3 $L_{p}$ Blaschke-Minkowski homomorphisms and convolutions

4 The Shephard-type problem

Acknowledgements

Competing interests

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Abstract

Competing interests

1 Introduction

2 Notation and background material

3 L p Blaschke-Minkowski homomorphisms and convolutions

4 The Shephard-type problem

Acknowledgements

Competing interests

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3 $L_{p}$ Blaschke-Minkowski homomorphisms and convolutions