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volume 7, issue 1, article 19, 2006.

Received 18 May, 2005;

accepted 22 November, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

ON BUSEMANN SURFACE AREA OF THE UNIT BALL IN MINKOWSKI SPACES

ZOKHRAB MUSTAFAEV

Department of Mathematics Ithaca College, Ithaca NY 14850 USA.

EMail:zmustafaev@ithaca.edu

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2000Victoria University ISSN (electronic): 1443-5756 156-05

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On Busemann Surface Area of the Unit Ball in Minkowski

Spaces Zokhrab Mustafaev

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J. Ineq. Pure and Appl. Math. 7(1) Art. 19, 2006

http://jipam.vu.edu.au

Abstract

For a givend-dimensional Minkowski space (finite dimensional Banach space) with unit ballB, one can define the concept of surface area in different ways whend≥3. There exist two well-known definitions of surface area: the Buse- mann definition and Holmes-Thompson definition of surface area. The purpose of this paper is to establish lower bounds for the surface area of the unit ball in a d-dimensional Minkowski space in case of Busemann’s definition, whend≥3.

2000 Mathematics Subject Classification:52A40, 46B20.

Key words: Busemann surface area, Cross-section measure, Isoperimetrix, Inter- section body, Mixed volumes, Projection body.

I would like to thank the referee for his comments and suggestions.

1. Introduction

It was shown by Goł¸ab (see [11] for details of this theorem) that in a two- dimensional Minkowski space the surface area of the the unit ball lies between 6 and 8 where the extreme values are attained if and only if the unit ball is a regular hexagon and a parallelogram, respectively. Recall that in a two-dimensional Minkowski space the surface area is defined by the induced norm of this space.

One can also raise the following question: “What are the extremal values of the surface area of the unit ball in a d-dimensional Minkowski space, when d ≥ 3?” To answer this question, first the notion of surface area needs to be defined, since the norm is no longer sufficient to define the surface area, when d ≥3. Various definitions of surface area were explored in higher dimensional Minkowski spaces (see [2,3,4,12,13]).

One of the definitions of surface area was given by Busemann in his pa- pers [1, 2, 3]. In [4], Busemann and Petty investigated this Busemann defini- tion of surface area for the unit ball when d ≥ 3. They proved that if B is the unit ball of a d-dimensional Minkowski space M^d = (R^d, || · ||), then its Busemann surface area ν_B(∂B) is at most 2d_d−1, and is equal to 2d_d−1 if and only if B is a parallelotope. Here _d stands for the volume of the standard d-dimensional Euclidean unit ball. They also raised the following question: “What is the extremum value for the lower bounds of this surface area?”

There have been obtained some lower bounds (not sharp) for this surface area of the unit ball in d−dimensional Minkowski spaces. In [12] (see also [13]), Thompson showed thatν_B(∂B)≥2d−1, andν_B(∂B)≥(d_d)

m_d ²_d

1/d

, where m_d := min{λ(B)λ(B^◦) : B a centered symmetric convex body inR^d}. In [12], Thompson also conjectured that for d > 3 the quantityνB(∂B) is min-

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imal for an ellipsoid.

One goal of this paper is to establish some lower bounds onν_B(∂B) when d ≥ 3. We will also prove that Thompson’s conjecture is valid when the unit ball possesses a certain property.

Furthermore, we shall show that in general Busemann’s intersection inequality cannot be strengthened to

λ^d−1(K)λ((IK)^◦)≥ _d

_d−1 d

in R^d. Namely, we present a counterexample to this inequality in R³. This result shows that the “duality” resemblance between projection and intersection inequalities does not always hold (cf. Petty’s projection inequality in Section 2).

We shall also show the relationship between the Busemann definition of surface area and cross-section measures.

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2. Definitions and Notations

One can find all these notions in the books of Gardner [5] and Thompson [13].

Recall that a convex bodyKis a compact, convex set with nonempty interior, and that K is said to be centered if it is centrally symmetric with respect to the origin 0of R^d. As usual, we denote by S^d−1 the standard Euclidean unit sphere inR^d. We writeλ_ifor ani-dimensional Lebesgue measure inR^d, where 1 ≤i≤d, and instead ofλdwe simply writeλ. Ifu∈ S^d−1, we denote byu^⊥ the (d−1)-dimensional subspace orthogonal to u, and by l_u the line through the origin parallel tou.

For a convex bodyKinR^d, we define the polar bodyK^◦ ofK by K^◦ ={y∈R^d:hx, yi ≤1, x∈K}.

We identifyR^dand its dual space R^d∗ by using the standard basis. In that case,λ_i andλ^∗_i coincide inR^d.

IfK₁ andK₂ are convex bodies inX, and α_i ≥0, i = 1,2, then the linear combination (for α1 = α2 = 1 the Minkowski sum) of these convex bodies is defined by

α₁K₁+α₂K₂ :={x:x=α₁x₁+α₂x₂, x_i ∈K_i}.

It is easy to show that the linear combination of convex bodies is itself a convex body.

IfK is a convex body inR^d, then the support functionh_K ofK is defined by

hK(u) = sup{hu, yi:y∈K}, u∈S^d−1,

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giving the distance from 0 to the supporting hyperplane of K with the outward normalu. Note thatK1 ⊂K2 if and only ifhK1 ≤hK2 for anyu∈S^d−1.

It turns out that every support function is sublinear, and conversely that every sublinear function is the support function of some convex set (see [13, p. 52]).

If0∈K, then the radial function ofK,ρ_K(u), is defined by ρ_K(u) = max{α ≥0 :αu ∈K}, u∈S^d−1,

giving the distance from 0 to l_u∩∂K in the directionu. Note thatK₁ ⊂ K₂if and only ifρ_K₁ ≤ρ_K₂ for anyu ∈S^d−1. Both functions have the property that forα₁, α₂ ≥0

h_α₁_K₁_+α₂_K₂(u) =α₁h_K₁(u) +α₂h_K₂(u), ρ_α₁_K₁_+α₂_K₂(u)≥α₁ρ_K₁(u) +α₂ρ_K₂(u) for any directionu.

We mention the relation

(2.1) ρK^◦(u) = 1

h_K(u), u∈S^d−1,

between the support function of a convex body K and the inverse of the radial function ofK^◦.

For convex bodies K1, ..., Kn−1, Kn in R^d we denote by V(K1, . . . , Kn) their mixed volume, defined by

V(K₁, ..., K_n) = 1 d

Z

S^d−1

h_K_ndS(K₁, ..., Kn−1, u)

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withdS(K₁, ..., Kn−1,·)as the mixed surface area element ofK₁,...,Kn−1. Note that we haveV(K₁, K₂, ..., K_n)≤V(L₁, K₂, ..., K_n)ifK₁ ⊂L₁, that V(αK₁, ..., K_n) = αV(K₁, ..., K_n),ifα ≥0and thatV(K, K, ..., K) =λ(K).

Furthermore, we will writeV(K[d−1], L)instead ofV(K, K, ..., K

| {z }

d−1

, L).

Minkowski’s inequality for mixed volumes states that ifK₁ andK₂ are convex bodies inR^d, then

V^d(K₁[d−1], K₂)≥λ^d−1(K₁)λ(K₂) with equality if and only ifK₁andK₂ are homothetic.

IfK₂ is the standard unit ball inR^d, then this inequality becomes the standard isoperimetric inequality.

One of the fundamental theorems on convex bodies refers to the Blaschke- Santalo inequality and states that ifK is a symmetric convex body inR^d, then

λ(K)λ(K^◦)≤²_d with equality if and only ifKis an ellipsoid.

The sharp lower bound is known only for zonoids. It is called the Mahler- Reisner Theorem which states that ifKis a zonoid inR^d, then

4^d

d! ≤λ(K)λ^∗(K^◦) with equality if and only ifKis a parallelotope.

Recall that zonoids are the limits of zonotopes with respect to the Hausdorff metric, and zonotopes are finite Minkowski sums of centered line segments.

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For a convex bodyK inR^d andu ∈ S^d−1 we denote by λd−1(K|u^⊥)the (d−1)-dimensional volume of the projection ofKonto a hyperplane orthogonal to u. Recall that λd−1(K|u^⊥) is called the (d−1)-dimensional outer cross- section measure or brightness ofK atu.

The projection body ΠK of a convex body K inR^d is defined as the body whose support function is given by

hΠK(u) = lim

ε→0

λ(K+ε[u])−λ(K)

ε =λd−1(K |u^⊥), where[u]is the line segment joining−^u₂ to ^u₂.

Note thatΠK = Π(−K), and that a projection body is a centered zonoid. If K₁ andK₂ are centered convex bodies inR^dandΠK₁ = ΠK₂, thenK₁ =K₂.

IfK is a convex body inR^d, then 2d

d

d^−d≤λ^d−1(K)λ((ΠK)^◦)≤ _d

d−1

d

with equality on the right side if and only ifKis an ellipsoid, and with equality on the left side if and only ifKis a simplex.

The right side of this inequality is called the Petty projection inequality, and the left side was established by Zhang (see [5]).

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3. Surface Area and Isoperimetrix

Let (R^d, || · ||) = M^d be a d-dimensional real normed linear space, i.e., a Minkowski space with unit ball B which is a centered convex body. The unit sphere ofM^dis the boundary of the unit ball and denoted by∂B.

A Minkowski spaceM^d possesses a Haar measure ν (orν_B if we need to emphasize the norm), and this measure is unique up to multiplication of the Lebesgue measure by a constant, i.e.,

ν =σ_Bλ.

It turns out that it is not as easy a problem to choose a right multipleσ as it seems. These two measuresν andλhave to coincide in the standard Euclidean space.

Definition 3.1. IfK is a convex body inR^d, then thed-dimensional Busemann volume ofK is defined by

ν_B(K) = _d

λ(B)λ(K), i.e., σ_B = _d λ(B).

Note that these definitions coincide with the standard notion of volume if the space is Euclidean, and thatν_B(B) = _d.

LetM be a surface inR^dwith the property that at each point xofM there is a unique tangent hyperplane, and that u_x is the unit normal vector to this hyperplane atx. Then the Minkowski surface area ofM is defined by

ν_B(M) :=

Z

M

σ_B(u_x)dS(x).

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For the Busemann surface area,σ_B(u)is defined by σ_B(u) = _d−1

λ(B ∩u^⊥).

The functionσ(u)can be extended homogeneously to the whole ofM^d, and it turns out that this extended function is convex (see [4], or [5]). Thus, this extended function σ is the support function of some convex body in R^d. We denote this convex body by TB, therefore if K is a convex body inM^d, then Minkowski’s surface area ofKcan also be defined by

(3.1) ν_B(∂K) =dV(K[d−1], T_B).

We deduce thatν_B(∂T_B) = dλ(T_B).

From Minkowski’s inequality for mixed volumes one can see thatT_B plays a central role regarding the solution of the isoperimetric problem in Minkowski spaces.

Among the homothetic images ofT_Bwe want to specify a unique one, called the isoperimetrixTˆ_B, determined byν_B(∂Tˆ_B) =dν_B( ˆT_B).

Proposition 3.1. IfB is the unit ball ofM^dandTˆ_B = ^λ(B)

d T_B, then ν_B(∂Tˆ_B) = dν_B( ˆT_B).

Proof. We use properties of the surface area, and straight calculation to obtain

ν_B(∂Tˆ_B) = λ^d−1(B)

^d−1_d ν_B(∂T_B)

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=dλ^d−1(B)

^d−1_d λ(T_B) = d _d

λ(B)λ( ˆT_B) =dν_B( ˆT_B).

Now we define the inner and outer radius of a convex body in a Minkowski space. Note that in Minkowski geometry these two notions are used with different meanings (see [11], [13]). As in [13], here these notions are defined by using the isoperimetrix.

Definition 3.2. If K is a convex body in R^d, the inner radius of K, r(K), is defined by

r(K) = max{α :∃x∈M^dwithαTˆ_B ⊆K+x}, and the outer radius ofK,R(K), is defined by

R(K) = min{α :∃x∈M^dwithαTˆ_B ⊇K+x}.

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4. The Intersection Body

We know that σB(f) = _λ(B∩u^d−1⊥) is a convex function and the support function ofT_B. Since the support function is the inverse of the radial function, we have that

ρ(u) =σ_B⁻¹(u) = ⁻¹_d−1λ(B∩u^⊥) is the radial function ofT_B^◦.

The intersection body ofK is a convex body whose radial function isλ(K∩ u^⊥)in a given directionu, and we denote it byIK(see [7] for more about intersection bodies). We can also rewrite the solution of the isoperimetric problem TBas

(4.1) T_B =d−1(IB)^◦.

One can see thatT_αB =α^1−dT_B forα≥0.

There is an important relationship between the volume of a convex body and the volume of its intersection body. It is called Busemann’s intersection inequality which states that ifK is a convex body inR^d, then

λ(IK)≤ d−1

_d d

²_dλ^d−1(K) with equality if and only ifKis a centered ellipsoid (see [5]).

SettingK = B in Busemann’s intersection inequality and using (4.1), we can rewrite this inequality as

(4.2) λ(T_B^◦)^d_d≤²_dλ^d−1(B).

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It turns out that ifK is a convex body inX with0as an interior point, then

(4.3) IK ⊆ΠK,

with equality if and only ifKis a centered ellipsoid (see [8]).

Recall that the intersection body of a centeredd-dimensional ellipsoidE is a centered ellipsoid, i.e., more precisely we have

IE = d−1λ(E) _d E.

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5. Some Lower Bounds on the Surface Area of the Unit Ball

As we mentioned in the introduction, the reasonable question is to ask how large and how small the surface area of the unit ball of M^d for the Busemann definition can be. In [4] Busemann and Petty showed that ifBis the unit ball of ad-dimentional Minkowski spaceM^d, then

ν_B(∂B)≤2dd−1

with equality if and only ifBis a parallelotope.

In this section we establish lower bounds for the Busemann surface area of the unit ball in ad-dimensional Minkowski space whend≥3.

Theorem 5.1. IfB is the unit ball of ad−dimensional Minkowski spaceM^d, then

νB(∂B)≥d−1

2d d

_d¹ .

Proof. SinceT_B =d−1(IB)^◦ ⊇d−1(ΠB)^◦, we get by Zhang’s inequality λ(T_B)≥^d_d−1λ((ΠB)^◦)≥

2d d

d^−d^d_d−1λ^1−d(B).

Therefore

d^dλ^d−1(B)λ(TB)≥ 2d

d

^d_d−1.

From Minkowski’s inequality it follows thatν_B^d(∂B)≥d^dλ^d−1(B)λ(T_B).Hence the result follows.

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We note that ^2d_d

≥2^d.

Theorem 5.2. If B is the unit ball of a d-dimensional Minkowski space M^d, then

ν_B(∂B)≥d_d

λ(T_B)λ(T_B^◦) ²_d

¹_d

with equality if and only ifBis an ellipsoid.

Proof. It follows from Busemann’s intersection inequality that λ(T_B^◦)≤(²_d/^d_d)λ^d−1(B).

Therefore

λ(T_B)λ(T_B^◦)≤(²_d/^d_d)λ^d−1(B)λ(T_B).

Using Minkowski’s inequality we get ν_B^d(∂B)

d^d^d_d ²_d≥λ(T_B)λ(T_B^◦).

Hence the inequality follows, and one can also see that equality holds if and only ifB is an ellipsoid.

Let us define µT_B(TB) = ^λ(T^B^)λ(T^B^◦⁾

d , i.e., the Holmes-Thompson definition of volume for T_B (see [6] or [13]) in a d-dimensional Minkowski space (R^d, T_B).

It follows from the Blaschke-Santalo inequality that _d

λ(TB)λ(T_B^◦) ²_d

¹_d

≥µ_T_B(T_B)

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with equality if and only ifBis an ellipsoid.

We obtain the following.

Corollary 5.3. If B is the unit ball of ad-dimensional Minkowski spaceM^d, then

ν_B(∂B)≥dµ_T_B(T_B), with equality if and only ifBis an ellipsoid.

We show that Thompson’s conjecture is valid when the unit ball possesses a certain property.

Theorem 5.4. IfB is the unit ball ofM^dwith an outer radius ofR(B), then

ν_B(∂B)≥ dd

R ,

with equality if and only ifB =R(B) ˆT_B.

Proof. SinceTˆBis the solution of the isoperimetric problem, we have ν_B^d(∂B)

ν_B^d−1(B) ≥ ν_B^d(∂Tˆ_B)

ν_B^d−1( ˆT_B) =d^dν_B( ˆT_B)≥ d^d

R^dν_B(B).

Hence the result follows, since ν_B(B) = _d. Obviously, if equality holds, then we getB =R(B) ˆT_B.IfB =R(B) ˆT_B,then we have

ν_B(∂B) =R^d−1ν_B(∂Tˆ_B) = d

RR^dν_B( ˆT_B) = d

Rν_B(B).

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Corollary 5.5. IfB is the unit ball of a d−dimensional Minkowski spaceM^d such thatR(B)≤1, then

ν_B(∂B)≥d_d,

with equality if and only ifB = ˆTB.

Proof. The inequality part and the implication follow from Theorem5.4.

Now assume thatR(B)≤1andν_B(∂B) = d_d.ThenB ⊆Tˆ_B, and d^d^d_d=d^dV^d(B[d−1], T_B)≥d^dλ^d−1(B)λ(T_B).

This gives us that λ(B) ≥ λ( ˆT_B). Henceλ(B) = λ( ˆT_B), and this is the case whenB = ˆT_B.

In [12], Thompson showed that if the unit ball is an affine regular rombic dodecahedron inR³,thenνB(∂B) =dd = 4π.Therefore, for a rombic dodecahedron in R³ eitherB = ˆT_B orR(B) > 1. The first one cannot be the case, since ifB is a rombic dodecahedron, then the facets of(IB)^◦ become “round”

(cf. [13, p. 153]).

Corollary 5.6. IfR(B)is the outer radius of the unit ball ofBin ad-dimensional Minkowski spaceM^d, then

R(B)≥ _d 2d−1

.

Proof. The result follows from the fact that ν_B(∂B) ≤ 2dd−1 and Theorem 5.4.

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In [9], it was proved that R(B) ≤ ₂^d^d

d−1 with equality if and only ifB is a parallelotope.

Theorem 5.7. If B is the unit ball of a d-dimensional Minkowski space M^d such thatλ( ˆT_B)≥λ(B),then

νB(∂B)≥dd,

with equality if and only ifB = ˆT_B. Proof. We can rewriteλ( ˆT_B)≥λ(B)as

λ^d−1(B)λ(T_B)≥^d_d. This gives us

ν_B^d(∂B) =d^dV^d(B[d−1], T_B)≥d^dλ^d−1(B)λ(T_B)≥d^d^d_d.

Hence the result follows. Obviously, if B = ˆT_B, then ν_B(∂B) = d_d. If νB(∂B) = dd, then it follows from Minkowski’s inequality that B and TB

must be homothetic. Thereforeλ( ˆT_B) = λ(B), and this is the case whenB = Tˆ_B.

From Theorem5.7it follows that ifBis a rombic dodecahedron inM^d, then λ( ˆT_B)< λ(B).

In [10] it was conjectured that if Iˆ_B is the isoperimetrix for the Holmes- Thompson definition in ad-dimensional Minkowski spaceM^d, then

λ( ˆI_B)≥λ(B) with equality if and only ifBis an ellipsoid.

Therefore, ifB is a rombic dodecahedron inR³, thenλ( ˆIB)> λ( ˆTB).

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Problem 1. Ifr(B)is the inner radius of the unit ballB for the isoperimetrix TˆB, is it then true that

r(B)≤1 with equality if and only ifBis an ellipsoid?

The answer of this question will tell us whether there exists a unit ball such that Tˆ_B ⊆ B. For the Holmes-Thompson definition of the isoperimetrix Iˆ_B, r(B)≤1holds with equality if and only ifB is an ellipsoid (see [10] or [13]).

In [13] (Problem 7.4.3, or p. 245) A.C. Thompson asked whether Buse- mann’s intersection inequality can be strengthened to

λ^d−1(K)λ((IK)^◦)≥ d

d−1

d

.

It is easy to show that equality holds for an ellipsoid. SettingK =B, we get λ( ˆT_B)≥λ(B).

As we have shown, the last inequality does not hold when B is a rombic dodecahedron inM³.

Now we show the relationship between cross-section measures and the Buse- mann definition of surface area.

Proposition 5.8. If the unit ballB ofM^dsatisfies λd−1(B∩u^⊥)λ₁(B|l_u)

λ(B) ≤ 2d−1

_d for eachu∈S^d−1, then

ν_B(∂B)≥d_d.

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Proof. It follows from the hypothesis of the proposition that for anyu∈S^d−1

ρ_IB(u)h_B(u)≤ d−1

_d λ(B).

Using (2.1), we get_dh_B(u)≤λ(B)h_T_B(u)for each direction, and therefore dB ⊆TBλ(B).

Hence the result follows from properties of mixed volumes and (3.1).

Problem 2.

a) Does there exist a centered convex bodyK inR^dsuch that λd−1(K∩u^⊥)λ₁(K|l_u)

λ(K) > 2d−1

_d

for eachu∈S^d−1?

b) Is it true that for a centered convex bodyK inR^d λd−1(K∩u^⊥)λ₁(K|l_u)

λ(K) = 2d−1

_d holds for eachu∈S^d−1only whenK is an ellipsoid?

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References

[1] H. BUSEMANN, The isoperimetric problem in the Minkowski plane, Amer. J. Math., 69 (1947), 863–871.

[2] H. BUSEMANN, The isoperimetric problem for Minkowski area, Amer. J.

Math., 71 (1949), 743–762.

[3] H. BUSEMANN, The foundations of Minkowskian geometry, Comment.

Math. Helv., 24 (1950), 156–187.

[4] H. BUSEMANN AND C.M. PETTY, Problems on convex bodies, Math.

Scand., 4 (1956), 88–94.

[5] R.J. GARDNER, Geometric Tomography, Encyclopedia of Mathematics and Its Applications 54, Cambridge Univ. Press, New York (1995).

[6] R.D. HOLMESANDA.C. THOMPSON,N-dimensional area and content in Minkowski spaces, Pacific J. Math., 85 (1979), 77–110.

[7] E. LUTWAK, Intersection bodies and dual mixed volumes, Adv. Math., 71 (1988), 232–261.

[8] H. MARTINI, On inner quermasses of convex bodies, Arch. Math., 52 (1989) 402–406.

[9] H. MARTINI AND Z. MUSTAFAEV, Some application of cross-section measures in Minkowski spaces, Period. Math. Hungar., to appear.

[10] Z. MUSTAFAEV, Some isoperimetric inequalities for the Holmes- Thompson definitions of volume and surface area in Minkowski spaces,

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J. Inequal. in Pure and Appl. Math., 5(1) (2004), Art. 17. [ONLINE:

http://jipam.vu.edu.au/article.php?sid=369].

[11] J.J. SCHÄFFER, Geometry of Spheres in Normed Spaces, Dekker, Basel 1976.

[12] A.C. THOMPSON, Applications of various inequalities to Minkowski ge- ometry, Geom. Dedicata, 46 (1993), 215–231.

[13] A.C. THOMPSON, Minkowski Geometry, Encyclopedia of Mathematics 63, Cambridge Univ. Press, 1996.