This thesis deals with the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies. The polar Orlicz-Minkowski problems are introduced and the solvability of such problems is discussed under different conditions. In particular, under certain condition on ϕ, the existence of a solution is proved for a nonzero finite measure µ on Sⁿ⁻¹ which is not concentrated on any hemisphere of Sⁿ⁻¹. The existence of the p-capacitary Orlicz-Petty bodies is also established. The Orlicz and L_q geominimal capacities with respect to K0 and S0 are proposed and their properties, such as invariance under orthogonal matrices, isoperimetric type inequalities and cyclic type inequalities are provided as well.

ii

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Acknowledgements

First and foremost, I would like to express my deepest gratitude to my thesis advisor Professor Deping Ye of Department of Mathematics and Statistics at Memorial Uni- versity of Newfoundland for his instructions of my thesis. He has patiently clarified every difficult spot in the thesis to me and given his support until the last second of my writing. Without his constant guidance, this thesis would never have been successfully completed.

I would also like to give my thanks to Dr. Baocheng Zhu, and I am particularly grateful for his valuable suggestions on this thesis. Thanks would also go to my graduate classmates who provided great help during my writing process: Shaoxiong Hou, Sudan Xing and Han Hong. I am so pleased to study with them.

Last but not the least, I would like to thank my parents for their continuous support and encouragement during my master’s study. Thank you.

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4.1 The nonhomogeneous and homogeneous Orlicz mixed p-capacities . . . 42 4.2 The p-capacitary Orlicz-Petty bodies . . . 49 4.3 The p-capacitary Orlicz-Petty bodies for multiple convex bodies . . . . 60 5 The Orlicz and L_q geominimal p-capacities 67 5.1 The Orlicz geominimal p-capacity . . . 67 5.2 The L_q geominimal p-capacity . . . 74 5.3 The mixed L_q geominimal p-capacity . . . 83

Bibliography 95

iv

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Chapter 1 Introduction

LetK0 denote the set of all convex bodies in Rⁿ with the origin o in their interiors, i.e., K ∈ K0 is a convex compact subset of Rⁿ such that o ∈ intK, the interior of K. The mixed volume is one of the central concepts in the Brunn-Minkowski theory of convex bodies. It comes naturally from the combination of Minkowski sum and volume in Rⁿ and specifically, the mixed volume ofK, L∈K0,denoted by V1(K, L), is the variation of the volume of K with respect toL, i.e.,

nV₁(K, L) = lim

→0⁺

V(K+L)−V(K)

.

The mixed volume ofK, L∈K0 (see e.g. [20, 57]), can also be formulated by

V₁(K, L) = 1 n

Z

Sⁿ⁻¹

h_L(u)dS(K, u),

whereS(K,·) is the surface area measure of K (see e.g. [1, 16]) and hLis the support function of L defined on Sⁿ⁻¹, the unit sphere of Rⁿ (see Chapter 2 for more details on the notations). The mixed volume plays fundamental roles in many important objects in convex geometry. For instance, the classical Minkowski inequality (see e.g.

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[17, 20, 57]) states that, for allK, L∈K0,

V₁(K, L)≥ |K|ⁿ⁻¹ⁿ |L|¹ⁿ (1.1)

with equality if and only if K and L are homothetic of each other, i.e. there exist a constant λ > 0 and x₀ ∈ Rⁿ such that K = λL+x₀. Hereafter |K| refers to the volume of K ∈ K0 and ω_n = |B₂ⁿ| denotes the volume of the unit ball B₂ⁿ in Rⁿ. In fact, the classical Minkowski inequality (1.1) implies that: for anyK ∈K0 given, the following optimization problem

inf (1

n Z

Sⁿ⁻¹

h_L(u)dS(K, u) : L∈K0 and |L|=ω_n )

(1.2)

has a unique solution (up to a translation).

The classical Minkowski problem is a fundamental problem in convex geometry and it has inspired a lot of problems with a similar nature, such asL_p Minkowski and Orlicz Minkowski problems. The classical Minkowski problem asks the necessary and sufficient conditions for a Borel measure µ on Sⁿ⁻¹ such that there exists a convex body K ∈ K0 with dS(K,·) = dµ. It has been proved in, e.g. [52, 53], that there exists a unique convex body K (up to a translation) such that dS(K,·) =dµ,if µis not concentrated on any hemisphere ofSⁿ⁻¹ and has the centroid at the origin, i.e.,

Z

Sⁿ⁻¹

hξ, ui₊dµ(u)>0 for any ξ∈Sⁿ⁻¹ and Z

Sⁿ⁻¹

udµ(u) =o,

wherehξ, ui+ = max{hξ, ui,0}.To solve the classical Minkowski problem is equivalent to find an optimizer for the following optimization problem:

inf (1

n Z

Sⁿ⁻¹

h_Ldµ: L∈K0 and |L|=ω_n )

. (1.3)

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Note that (1.3) is obtained by replacing the surface area measureS(K,·) byµin (1.2).

Another important concept in convex geometry closely related to the mixed volume is the classical geominimal surface area, which is denoted by G(·) and is defined by [56]: for any K ∈K0,

G(K) = inf (Z

Sⁿ⁻¹

h_L(u)dS(K, u) : L∈K0 and |L^◦|=ω_n )

, (1.4)

whereL^◦ denotes the polar body of L, i.e., L^◦ ={y∈Rⁿ:hx, yi ≤1 for anyx∈L}.

It has been proved in [56] that there exists a (unique) convex body T₁K, which is called the Petty body ofK, such that

G(K) =nV₁(K, T₁K) = Z

Sⁿ⁻¹

h_T₁_K(u)dS(K, u) and |(T₁K)^◦|=ω_n.

In view of (1.3) and (1.4), one sees that the main difference is to replace L in (1.3) by L^◦ in (1.4). Consequently, one may call the optimization problem in (1.4) the polar Minkowski problem. We would like to mention that both (1.3) and (1.4) are the key ingredients in the development of the Brunn-Minkowski theory of convex bodies.

Indeed, a variation problem of (1.4) [42]

inf (

V₁(K, L^◦) = 1 n

Z

Sⁿ⁻¹

ρ⁻¹_L (u)dS(K, u) :L∈S0 and |L|=ω_n )

,

where ρ_L : Sⁿ⁻¹ → (0,∞) is the radial function of a star body L ∈ S0, provides an equivalent formula for the classical affine area [4] and plays arguably more important roles in the Brunn-Minkowski theory of convex bodies, due to its applications in, such as, the theory of valuation (see e.g. [2, 3, 38]) and the approximation of convex bodies by polytopes (see e.g. [19, 40, 59]).

Forp ∈R but p6= 0,1,the L_p mixed volume of K, L∈ K0 (see e.g. [43, 68]) can

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be defined as

V_p(K, L) = 1 n

Z

Sⁿ⁻¹

h_L(u) h_K(u)

p

h_K(u)dS(K, u). (1.5)

The Lp mixed volume plays fundamental roles in the development of the Lp Brunn- Minkowski theory of convex bodies (see e.g. [6, 7, 22, 44, 45, 48, 51, 59]). Analogous to (1.1), one has the L_p Minkowski inequality [43]: for all p >1 and all K, L∈K0,

V_p(K, L)≥ |K|^n−pⁿ |L|^pⁿ (1.6)

with equality if and only ifK and L are dilates of each other, i.e., K =λL for some λ >0. Again the L_p Minkowski inequality (1.6) implies that: for any fixed K ∈ K0

and p >1, the following optimization problem

inf (Z

Sⁿ⁻¹

h_L(u) h_K(u)

p

hK(u)dS(K, u) :L∈K0 and |L|=ωn

)

(1.7)

has a unique solution.

Related to (1.7) is the L_p Minkowski problem: for p ∈ R and p6= 0, under what condition on a given nonzero finite measureµdefined onSⁿ⁻¹,there is a convex body K (ideally with the origin o in its interior) such that h^p−1_K dµ = dS(K,·)? The L_p Minkowski problem is a popular problem in geometry and has attracted considerable attention (see e.g. [8, 9, 29, 32, 43, 47, 60, 76, 77, 78]). Solutions to theL_p Minkowski problems have fundamental applications in, for instance, establishing the L_p Sobolev type inequalities (see e.g. [10, 23, 46, 72]).

Replacing L by L^◦ in (1.7), one can ask the following problem: for p > 1, find a convex body to solve

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inf (Z

Sⁿ⁻¹

h_L(u) h_K(u)

p

h_K(u)dS(K, u) :L∈K0 and |L^◦|=ω_n )

. (1.8)

In fact, Lutwak in [44] showed that there exists a unique convex bodyT_pK ∈K0, the L_p Petty body of K, such that |(T_pK)^◦|=ω_n and

nV_p(K, T_pK) = inf (Z

Sⁿ⁻¹

h_L(u) h_K(u)

p

.

The quantity G_p(K) = nV_p(K, T_pK) for p > 1 is called the L_p geominimal surface area of K [44]. Recently, Ye extended the L_p geominimal surface area of K to all 06=p∈R [68] and Zhu, Hong and Ye showed the existence of the L_p Petty body for p >0 [74]. Replacing K0 by S0 in (1.7), one gets,

Ω_p(K) = inf (Z

Sⁿ⁻¹

1 h_K(u)ρ_L(u)

p

h_K(u)dS(K, u) :L∈S0 and |L|=ω_n )

.

In literature, Ω_p(K) is called theL_p affine surface area and has equivalent convenient integral formulas (see e.g. [31, 44, 51, 58, 59]). It is well known that the L_p affine surface area has many applications in, such as, the valuation theory, the approximation of convex bodies by polytopes, the f-divergence of convex bodies and the L_p affine isoperimetric inequalities (see e.g. [19, 28, 33, 39, 40, 55, 59, 61, 62, 63, 73]).

Extension from the L₁ and L_p Brunn-Minkowski theories to the Orlicz theory is rather dedicated and involves nonhomogeneous functions. In view of (1.2)-(1.8), a major task is to get the “right” formula of the Orlicz mixed volume. In the Orlicz theory, there are at least 3 different ways to define the Orlicz mixed volume and each of them has their own advantages. These Orlicz mixed volumes are given as follows:

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forK, L∈K0, and φ, ϕ: (0,∞)→(0,∞) continuous functions,

V_ϕ(K, L) = 1 n

Z

Sⁿ⁻¹

ϕh_L(u) h_K(u)

h_K(u)dS(K, u), (1.9) V_ϕ,φ(K, L) = 1

n Z

Sⁿ⁻¹

ϕ(h_L(u))

φ(hK(u))dS(K, u), (1.10) and if in addition ϕ∈I,

Vb_ϕ(K, L) = inf (

λ >0 : Z

Sⁿ⁻¹

ϕn|K| ·h_L(u) λ·h_K(u)

h_K(u)dS(K, u)≤n|K|

)

, (1.11)

whereI refers to the set of continuous functions ϕ: (0,∞) →(0,∞) such that ϕis strictly increasing, lim_t→0⁺ϕ(t) = 0, ϕ(1) = 1 and limt→∞ϕ(t) = ∞. The L_p mixed volume defined by (1.5) is a special case of (1.9)-(1.11). For instance,

V_p(K, L) = V_ϕ(K, L) =V_ϕ,φ(K, L) =n^−p· |K|^1−p Vb_ϕ(K, L)p

if ϕ(t) = t^p and φ(t) = t^p−1. Note that V_ϕ(·,·) given by (1.9) and V_ϕ,φ(·,·) given by (1.10) can be obtained by the combination of volume and a family of linear Orlicz additions of K and L (see e.g. [18, 65, 74]). However, Vb_ϕ(·,·) seems not have a geometric interpretation. On the other hand, bothV_ϕ(·,·) andVb_ϕ(·,·) have the Orlicz- Minkowski inequalities [18, 65], which extend the L₁ and L_p Minkowski inequalities.

The Orlicz-Minkowski inequalities read: forK, L∈K0andϕbeing a convex function, then

V_ϕ(K, L)≥ |K| ·ϕ

|L|

|K|

_n¹!

, (1.12)

and if ϕ∈I,

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Vb_ϕ(K, L)≥n|K|ⁿ⁻¹ⁿ |L|ⁿ¹. (1.13) Equality holds if and only ifK andLare dilates, ifϕis also strictly convex. Also note that Vb_ϕ(·,·) has homogeneity [74] but V_ϕ(·,·) and V_ϕ,φ(·,·) do not have homogeneity.

The Orlicz-Minkowski inequalities (1.12) and (1.13) imply that, ifϕ∈I is strictly convex, the following problems

inf (Z

Sⁿ⁻¹

ϕ

h_L(u) h_K(u)

hK(u)dS(K, u) :L∈K0 and |L|=ωn

)

and infn

Vb_ϕ(K, L) :L∈K0 and |L|=ω_no

have unique solutions. On the other hand, it seems attractable to pose the Minkows- ki type problems related to Vϕ(·,·) and Vbϕ(·,·) similar to the Lp Minkowski problems. However, such Orlicz-Minkowski problems can be asked for the case related to V_ϕ,φ(·,·): under what condition on a finite measure µ on Sⁿ⁻¹ and on a continuous function ϕ : (0,∞) → (0,∞), there exists a convex body K such that dS(K,·) = c·ϕ(h_K)dµfor some positive constantc? Solutions of this Orlicz-Minkowski problems can be found in [21, 30, 37].

Ye [69] and Zhu, Hong and Ye [74] investigated the following optimization problems and gave a detailed study of the (homogeneous and nonhomogeneous) Orlicz geominimal surface areas G^orlicz_ϕ (·) and Gb^orlicz_ϕ (·): under certain conditions on ϕ, define

G^orlicz_ϕ (K) = inf (Z

Sⁿ⁻¹

ϕh_L(u) h_K(u)

,

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Gb^orlicz_ϕ (K) = inf (

Vb_ϕ(K, L) :L∈K0 and |L^◦|=ω_n )

.

Again, one can replace K0 by S0 to get (homogeneous and nonhomogeneous) Orlicz affine surface areas. In particular, Zhu, Hong and Ye [74] proved that, under certain conditions on ϕ, there exist convex bodies TϕK and TbϕK, called the Orlicz-Petty bodies ofK, such that |(T_ϕK)^◦|=|(Tb_ϕK)^◦|=ω_n,

G^orlicz_ϕ (K) = nV_ϕ(K, T_ϕK) and Gb^orlicz_ϕ (K) = Vb_ϕ(K,Tb_ϕK).

One can also see [71] for a special case.

In Chapter 3, we will study the polar Orlicz-Minkowski problems, which are the Orlicz settings of theL_p polar Minkowski problems: under what conditions on ϕand a finite nonzero measureµdefined on Sⁿ⁻¹,there exists a convex body M ∈K0 such that |M^◦|=ω_n and

Z

Sⁿ⁻¹

ϕ(h_M(u))dµ(u) = inf (Z

Sⁿ⁻¹

ϕ(h_L(u))dµ(u) :L∈K0 and |L^◦|=ω_n )

.

Our main result in Chapter 3 is summarized in the following theorem. Let Ω be the set of all finite positive Borel measures on Sⁿ⁻¹ that are not concentrated on any hemisphere ofSⁿ⁻¹.

Theorem 1.1. Let µ∈Ωand ϕ∈I. Then there exists a convex body M ∈K0 such that|M^◦|=ω_n and

Z

Sⁿ⁻¹

ϕ(h_M(u))dµ(u) = inf (Z

Sⁿ⁻¹

ϕ(h_L(u))dµ(u) :L∈K0 and |L^◦|=ω_n )

.

Moreover, if ϕ ∈ I is convex, then M is the unique solution to the polar Orlicz- Minkowski problem.

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In Chapter 4, we replaceV_ϕ(·,·) andVb_ϕ(·,·) in (1.9) and (1.11) by theirp-capacitary counterparts, and study the existence and continuity of thep-capacitary Orlicz-Petty bodies. Here for K, L ∈ K0, p ∈ (1, n) and ϕ : (0,∞) → (0,∞), the Orlicz mixed p-capacities of K and L are given by:

Cp,ϕ(K, L) = p−1 n−p

Z

Sⁿ⁻¹

ϕ

h_L(u) h_K(u)

hK(u)dµp(K, u), (1.14) Z

Sⁿ⁻¹

ϕ C_p(K)·h_L(u) Cb_p,ϕ(K, L)·h_K(u)

!

dµ^∗_p(K, u) = 1 for ϕ∈I, (1.15)

where µ_p(K,·) is the p-capacitary measure given by (2.10) and µ^∗_p(K,·) is the nor- malized p-capacitary measure on Sⁿ⁻¹ given by (2.15). Note that C_p,ϕ(·,·) can be obtained by the combination of thep-capacity and a family of linear Orlicz additions ofK andL[26]. Here for a compact setE ⊆Rⁿ,thep-capacity ofE (see e.g. [14, 15]), is defined by

C_p(E) = inf Z

Rⁿ

|∇f(x)|^pdx: f ∈C_c^∞(Rⁿ) and f(x)≥1 on x∈E

,

where C_c^∞(Rⁿ) denotes the set of all infinitely differentiable functions on Rⁿ with compact supports and ∇f denotes the gradient of f ∈ C_c^∞(Rⁿ). Again Cb_p,ϕ(·,·) has homogeneity aboutK and L (see e.g. [26] or Corollary 4.1), i.e.,

Cb_p,ϕ(sK, tL) = s^n−p−1·t·Cb_p,ϕ(K, L), for any s >0 and any t >0.

Related to C_p,ϕ(·,·) and Cb_p,ϕ(·,·), there are the p-capacitary Orlicz-Minkowski inequalities: forp∈(1, n), K, L∈K0 and ϕ∈I convex [26], one has,

C_p,ϕ(K, L)≥C_p(K)·ϕ

Cp(L) C_p(K)

_n−p¹ !

and Cb_p,ϕ(K, L)≥C_p(K)

Cp(L) C_p(K)

_n−p¹ ,

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with equality if and only if K and L are dilates, if ϕ is strictly convex. From these inequalities, we can get the following facts: for p∈(1, n) and ϕ∈I strictly convex,

infn

C_p,ϕ(K, L) :L∈K0 and |L|=ω_no , inf

n

Cbp,ϕ(K, L) :L∈K0 and |L|=ωn

o

have unique solutions. Analogous to the L_q and Orlicz Minkowski problems related toS(K,·),one can ask thep-capacitaryL_q and Orlicz Minkowski problems (i.e., with S(K,·) replaced by µ_p(K,·)). These problems have received extensive attention, see [13, 25, 26, 34, 35, 79] for more details.

Replacing L by L^◦, we will study the following problems: for p∈ (1, n), find the optimizers to the following optimization problems:

sup/infn

C_p,ϕ(K, L) :L∈K0 and |L^◦|=ω_no , sup/infn

Cb_p,ϕ(K, L) :L∈K0 and |L^◦|=ω_no .

Our main result in Chapter 4 is stated as follows:

Theorem 1.2. Let K ∈K0 be a convex body and ϕ∈I. (i) There exists a convex body M ∈K0 such that |M^◦|=ω_n and

C_p,ϕ(K, M) = infn

C_p,ϕ(K, L) :L∈K0 and |L^◦|=ω_no .

(ii) There exists a convex body Mc∈K0 such that |Mc^◦|=ω_n and

Cbp,ϕ(K,Mc) = inf n

Cbp,ϕ(K, L) :L∈K0 and |L^◦|=ωn

o .

In addition, ifϕ∈I is convex, then both M and Mcare unique.

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The convex bodies M and Mc in Theorem 1.2 are called the p-capacitary Orlicz- Petty bodies ofK. The continuity of thep-capacitary Orlicz-Petty bodies is provided in Theorem 4.2. In Chapter 5, we propose the Orlicz geominimal p-capacities of K and provide a detailed study on their properties. For instance, we define the homogeneous Orlicz geominimal p-capacity of K with respect to K0 for ϕ ∈ I by Gbϕ^orlicz(K) =Cb_p,ϕ(K,Mc),whereMcis the convex body given in Theorem 1.2. Proper- ties ofGbϕ^orlicz(K) are provided, such as the invariance under orthogonal matrices. In particular, we show the following inequality.

Theorem 1.3. LetK ∈K0 be a convex body with its Santal´o point or centroid at the origin and B_K be an origin symmetric ball defined by B_K =vrad(K)Bⁿ₂.

(i) If ϕ∈I0∪D0, then

Ac_p,ϕ^orlicz(K)

Ac_p,ϕ^orlicz(B_K) ≤ Gb_p,ϕ^orlicz(K)

Gbp,ϕ^orlicz(BK) ≤ Cp(K) C_p(B_K). Equality holds if K is an origin symmetric ball.

(ii) Ifϕ∈D1, then there exists a universal constant c > 0 such that

Ac_p,ϕ^orlicz(K)

Acp,ϕ^orlicz(B_K) ≥ Gb_p,ϕ^orlicz(K)

Gbp,ϕ^orlicz(BK) ≥c· C_p(K) C_p(B_K).

Similarly, we could define Gϕ^orlicz(K) =C_p,ϕ(K, M) where M is the convex body given in Theorem 1.2, and establish properties and inequalities similar to those for Gbϕ^orlicz(K).Special attention is paid on the case when ϕ(t) =t^q for −n6=q ∈R. Be- sides, we also investigate thep-capacitary Orlicz-Petty bodies ofKKK = (K₁,· · · , K_m), a vector of convex bodies, and establish analogous results for theL_q mixed geominimal p-capacity.

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Chapter 2 Preliminaries and Notations

A subset K ⊆ Rⁿ is said to be convex if λx+ (1−λ)y ∈ K for any λ ∈ [0,1] and x, y ∈ K. A convex body is a convex compact subset of Rⁿ with nonempty interior.

A convex bodyK is said to be origin-symmetric if−x∈K for anyx∈K. We useK and K0 ⊆K to denote the set of all convex bodies and the set of all convex bodies with the origin in their interiors, respectively. The Minkowski sum of K, L ∈ K , denoted byK+L, is defined by

K+L={x+y:x∈K, y∈L}.

Forλ∈R, the scalar product of λ and K, denoted by λK, is defined by

λK ={λx:x∈K}.

ForK ∈ K , |K| refers to the volume of K. In particular, ωn represents the volume of the unit ballB₂ⁿ ⊆Rⁿ. ForK ∈K , one can define the volume radius of K by

vrad(K) = |K|

ω_n ¹_n

.

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For an×nmatrixφ, detφrefers to the determinant ofφandφ^trefers to the transpose of φ. If detφ 6= 0, we use φ⁻¹ to denote the inverse of φ. By an ellipsoid E, we refer to a subset ofRⁿ given by

E =φB₂ⁿ+x₀ ={φx+x₀ :x∈B₂ⁿ},

whereφ is an invertiblen×nmatrix on Rⁿ and x₀ ∈Rⁿis some vector. Let O(n) be the set of alln×n matrices such thatφφ^t =φ^tφ=I_n, where I_n is the identity matrix onRⁿ.

The polar body K^◦ of K ∈K0 is defined by

K^◦ ={x∈Rⁿ :hx, yi ≤1 for any y∈K},

whereh·,·i is the standard inner product in Rⁿ. By K^◦◦, we mean the polar body of K^◦, and it is well known that K^◦◦ =K if K ∈ K0 [57, Theorem 1.6.1]. For K ∈K and z ∈intK, the polar body ofK with respect toz, denoted byK^z, is defined as

K^z = (K −z)^◦+z.

It has been proved in [50] that there exists a unique pointz₀ ∈intK, such that,

|K^z⁰|= inf{|K^z|: z ∈intK}.

This unique pointz₀ ∈intK, denoted by s(K), is called the Santal´o point of K. The famous Blaschke-Santal´o inequality can be stated as follows: for any K ∈K ,

|K| · |K^s(K)| ≤ω²_n (2.1)

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with equality if and only ifK is an ellipsoid. On the other hand, the inverse Santal´o inequality reads (see e.g. [5, 36, 54]): there exists a universal constantc >0 such that for any K ∈K ,

|K| · |K^s(K)| ≥cⁿω_n². (2.2) The support function of a nonempty convex compactK ⊆Rⁿ, h_K :Sⁿ⁻¹ →R, is defined by

h_K(u) = max

x∈K hx, ui for any u∈Sⁿ⁻¹,

where Sⁿ⁻¹ is the unit sphere. It can be easily checked that, for any λ ≥ 0 and K, L ∈ K , h_λK(u) = λh_K(u) and h_K+L(u) = h_K(u) + h_L(u) for any u ∈ Sⁿ⁻¹. A subset L ⊆Rⁿ is called a star-shaped set about the origin if for any x ∈ L, the line segment from the origino toxis contained inL. The radial function of a star-shaped setL about the origin o, ρ_L :Sⁿ⁻¹ →[0,∞), is defined by

ρL(u) = max{r ≥0 : ru ∈L} for any u∈Sⁿ⁻¹.

A star-shaped set L⊆Rⁿ about the origin o is called a star body about the origin o if the radial functionρ_L is positive and continuous on Sⁿ⁻¹. Denote by S0 the set of all star bodies about the origino. Obviously,K0 ⊆S0. It can be proved in [57] that for any K ∈K0,

ρ_K^◦(u) = 1

hK(u) and h_K^◦(u) = 1

ρK(u) for any u∈Sⁿ⁻¹. (2.3) ForK ∈S0, the following volume formula for |K| holds:

|K|= 1 n

Z

Sⁿ⁻¹

ρ_K(u)ⁿdσ(u), (2.4)

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whereσ(·) is the spherical measure on Sⁿ⁻¹. Associated to eachK ∈K0, the surface area measure (a positive Borel measure) onSⁿ⁻¹ of K, denoted byS(K,·),is defined by: for any measurable subsetA ⊆Sⁿ⁻¹,

S(K, A) = Z

ν⁻¹_K (A)

dH ⁿ⁻¹,

where ν_K⁻¹ : Sⁿ⁻¹ → ∂K is the inverse Gauss map [57] and Hⁿ⁻¹ is the (n− 1)- dimensional Hausdorff measure on∂K.If the measureS(K,·) is absolutely continuous with respect to the spherical measure σ(·), by the Radon-Nikodym theorem, there is a function f_K: Sⁿ⁻¹ → R, called the curvature function of K, such that, for any u∈Sⁿ⁻¹,

dS(K, u) =fK(u)dσ(u).

ByF0⁺, we mean the subset of K0 defined by:

F₀⁺ ={K ∈K0 :f_Kexists and is positively continuous on Sⁿ⁻¹}.

The Hausdorff distance betweenK, L∈K0, denoted by dH(K, L), is given by

d_H(K, L) = max

u∈Sⁿ⁻¹|h_K(u)−h_L(u)|.

In fact, the Hausdorff distance can be defined on two compact sets [57, (1.60)], i.e., for compact setsE, F ⊆Rⁿ,

d_H(E, F) = min{λ≥0 :E ⊆F +λB₂ⁿ and F ⊆E+λB₂ⁿ}.

For a sequence {Ki}^∞_i=1 ⊆K0 and a convex compact set K, Ki →K asi → ∞ with respect to the Hausdorff metric means that d_H(K_i, K)→0 as i→ ∞.

(20)

Denote by C(Sⁿ⁻¹) the set of all continuous functions on Sⁿ⁻¹. Let {µ_i}^∞_i=1 be a sequence of measures onSⁿ⁻¹ andµalso be a measure on Sⁿ⁻¹. We sayµ_i converges weakly toµif for any f ∈C(Sⁿ⁻¹),

i→∞lim Z

Sⁿ⁻¹

f dµ_i = Z

Sⁿ⁻¹

f dµ.

The following variation formula of weak convergence µ_i →µis often used.

Lemma 2.1. If a sequence of measures {µ_i}^∞_i=1 on Sⁿ⁻¹ converges weakly to a finite measureµonSⁿ⁻¹and a sequence of functions{f_i}^∞_i=1 ⊆C(Sⁿ⁻¹)converges uniformly to a function f ∈C(Sⁿ⁻¹), then

i→∞lim Z

Sⁿ⁻¹

fidµi = Z

Sⁿ⁻¹

f dµ.

Proof. The weak convergence of µ_i →µgives that for any f ∈C(Sⁿ⁻¹),

Z

Sⁿ⁻¹

f dµ_i− Z

Sⁿ⁻¹

f dµ

→0. (2.5)

In particular, whenf(u) = 1 for any u∈Sⁿ⁻¹, one gets

Z

Sⁿ⁻¹

1dµ_i− Z

Sⁿ⁻¹

1dµ

=

µ_i(Sⁿ⁻¹)−µ(Sⁿ⁻¹)

→0. (2.6)

Together withµ(Sⁿ⁻¹)<∞, one gets, for any i big enough, say i≥N₀,

µ_i(Sⁿ⁻¹)≤µ(Sⁿ⁻¹) + 1. (2.7)

On the other hand, the uniform convergence of f_i →f gives that

max

u∈Sⁿ⁻¹

f_i(u)−f(u) →0.

Table of contents

Abstract

Acknowledgements

Table of contents

Chapter 1 Introduction

Chapter 2

Preliminaries and Notations