Orlicz–John ellipsoids Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

Orlicz–John ellipsoids

Du Zou, Ge Xiong^∗

DepartmentofMathematics,ShanghaiUniversity,Shanghai,200444,PRChina

a r t i c l e i n f o a bs t r a c t

Article history:

Received3July2013 Accepted13July2014

Availableonline15August2014 CommunicatedbyErwinLutwak

MSC:

52A40

Keywords:

OrliczBrunn–Minkowskitheory LpJohnellipsoids

Isotropy

TheOrlicz–Johnellipsoids,whichareintheframeworkofthe boomingOrliczBrunn–Minkowskitheory,areintroducedfor thefirsttime.Itturnsoutthattheyaregeneralizationsofthe classical John ellipsoid and theevolved Lp John ellipsoids.

TheanalogofBall’svolume-ratioinequalityisestablishedfor thenewOrlicz–Johnellipsoids.Theconnection betweenthe isotropyofmeasuresandthecharacterizationofOrlicz–John ellipsoidsisdemonstrated.

© 2014ElsevierInc.All rights reserved.

1. Introduction

AfundamentaltoolinconvexgeometryandBanachspacegeometryisthewell-known Johnellipsoid,whichwasoriginallyintroducedbyFritzJohn[26].Foreachconvexbody (compact convex subset with nonempty interior) K in the Euclidean n-space Rⁿ, its John ellipsoid JK istheunique ellipsoidof maximal volumecontainedinK.For more information about the John ellipsoid, one can refer to [3,19,21,27] and the references within.

✩ Researchofthe authorswassupportedbyNSFCNo.11001163andInnovationProgramofShanghai MunicipalEducationCommissionNo.11YZ11.

* Correspondingauthor.

E-mailaddress:[email protected](G. Xiong).

http://dx.doi.org/10.1016/j.aim.2014.07.034 0001-8708/© 2014ElsevierInc.All rights reserved.

TheJohnellipsoid is withintheclassical Brunn–Minkowskitheoryand is extremely usefulinconvex geometry,Banachspace geometry andPDEs(see, e.g.,[1,2,16,28,47]).

One important result concerning the John ellipsoid is Ball’s volume-ratio inequality, whichstatesthat:ifK isanorigin-symmetricconvexbodyinRⁿ,then

|K|

|JK| ≤ 2ⁿ

ω_n, (1.1)

withequalityifandonlyifKisaparallelotope.Here,|· |denotesn-dimensionalvolume and ωn =π^n/2/Γ(1+ⁿ₂) denotesthevolume ofthe unitball, B, inRⁿ. Thefactthat thereisequalityin(1.1)onlyforparallelotopeswasestablishedbyBarthe[4].

In 2005, the classical John ellipsoid had evolved into the L_p John ellipsoids under theimpetusofLutwak,YangandZhang[38].Inretrospect,itisinterestingthatittook nearly adecade fortheLp Johnellipsoids to be discovered, after theemergenceof the Lp Brunn–Minkowskitheory initiatedby Lutwak [32,33]. Duringthe last two decades, theL_p Brunn–Minkowskitheoryhasachievedgreatdevelopmentsandexpandedrapidly (see,e.g.,[7–9,11,12,22–24,29–38,42,49,51–53,55–58]).

Suppose p ∈ (0,∞] and K is a convex body in Rⁿ with the origin in its interior.

Amongst all origin-symmetric ellipsoids E, the unique ellipsoid that solves the con- strainedmaximizationproblem

maxE

|E| ωn

¹_n

subject to Vp(K, E)≤1 iscalledtheLp Johnellipsoid [38]ofK anddenotedbyEpK. Here

Vp(K, E) =

Sⁿ⁻¹

hE

hK

p

dVK

¹_p

, 0< p <∞,

isthenormalizedLpmixedvolumeofKandE;Sⁿ⁻¹istheunitsphereinRⁿ;hKandhE

arethesupportfunctionsofK and E,respectively;VK is thenormalizedcone-volume measure of K. For p=∞, we define V_∞(K,E)= sup{h_E(u)/h_K(u) :u∈ suppV_K}. Notethatthecone-volumemeasurehasbeenappearedandinvestigatedwidelyinvarious contextsrecently(seee.g.,[5,7,8,18,24,25,30,31,43,45,52,53,57]).

Ingeneral, theLp John ellipsoid EpK is notcontained inK (exceptwhen p=∞).

However,when1≤p≤ ∞,ithas|E_pK|≤ |K|.IfKisanorigin-symmetricconvexbody inRⁿ and 0< p≤ ∞,thentheLp version[38]ofBall’svolume-ratioinequality

|K|

|EpK| ≤ 2ⁿ ωn

stillholds,with equalityifand onlyifK isaparallelotope.

The Lp John ellipsoids provide aunified treatmentfor several fundamental objects in convex geometry. If the John point of K, i.e., the center of JK, is at the origin, then E_∞K is precisely the classical John ellipsoid JK. The L₂ John ellipsoid E₂K is the new ellipsoid Γ₋₂K previously found by Lutwak, Yang and Zhang in [34], which is now called the LYZ ellipsoid and is in some sense dual to the Legendre ellipsoid of inertia in classical mechanics [41]. The L1 John ellipsoid E1K is the so-called Petty ellipsoid.The volume-normalized Pettyellipsoid is obtainedbyminimizingthe surface areaofKunderSL(n) transformationsofK.SeePetty[46]andalsoGiannopoulosand Papadimitrakis[17].

Beginning with the ground-breaking articles of Lutwak, Yang, Zhang and Harbel [22,39,40], amorewideextension of theLp Brunn–Minkowskitheory, called theOrlicz Brunn–Minkowskitheory,emergedoutthreeyearsago.Inthesearticles,thefundamental notions of the L_p projection body and the L_p centroid body wereextended to an Or- liczsetting (seealso[10,59]). Itrepresentsageneralizationof theLp Brunn–Minkowski theory,analogousto thewaythatOrliczspacesgeneralizeLpspaces[48].Very recently, oneessentialobstacleinthedevelopmentofOrliczBrunn–Minkowskitheory,whatisthe lackofanotioncorresponding toL_paddition,hasbeensmoothedbyGardner,Hugand Weil [14,15].

In viewof thefundamentalimportanceofthe Johnellipsoidinconvex geometry,we are tempted to consider the naturally posed problem in the booming Orlicz Brunn–

Minkowskitheory:whatistheOrliczextension oftheLp Johnellipsoid?Ourmaintask inthispaperisto demonstratethisexistenceofsuchanOrliczanalogue.

For this aim, weconsider convex ϕ: [0,∞)→[0,∞), thatis strictly increasingand satisfies ϕ(0) = 0. Forconvex bodies K,L inRⁿ with theorigin intheir interiors, the normalized OrliczmixedvolumeofK andL regardingϕ,Vϕ(K,L),isdefinedby

Vϕ(K, L) =ϕ⁻¹

Sⁿ⁻¹

ϕ h_L

hK

dVK

.

Inspired byLutwak,Yang andZhang’sworkonLp Johnellipsoids [38],wefocus on ProblemSϕ.SupposeKisaconvexbodyinRⁿwiththeorigininitsinterior.Findanel- lipsoidE,amongstallorigin-symmetricellipsoids,whichsolvesthefollowingconstrained maximizationproblem:

maxE |E| subject to V_ϕ(K, E)≤1.

In Section4,we provethatthereexists auniqueellipsoid whichsolvesProblemSϕ. ItiscalledtheOrlicz–Johnellipsoid ofK,anddenotedbyEϕK.Notethatifϕ(t)=t^p, 1≤p<∞,then theOrlicz–Johnellipsoid E_ϕK precisely turns outto be theL_p John ellipsoid EpK.

Animportantfeature onthefamily of Lp John ellipsoids isthatEpK is continuous inp∈(0,∞]. InSection5, weshow thattheOrlicz–Johnellipsoid EϕK is jointlycon- tinuous inϕ and K. In Section6, we provethat as p→ ∞, the Orlicz–Johnellipsoid Eϕ^pK approaches to E_∞K. This insight throwslight ona connectionbetween Orlicz–

JohnellipsoidsandtheclassicalJohnellipsoid.TheOrliczversionofBall’svolume-ratio inequalityisestablishedinSection7.Finally,weprovideacharacterizationoftheOrlicz–

Johnellipsoid,whichiscloselyrelatedto theisotropyofmeasures.

2. Preliminaries

For quickreference we recall somebasic results from the Brunn–Minkowski theory.

GoodreferencesareGardner[13],Gruber[20],Schneider[50],andThompson[54].

Thesettingwillbe Euclideann-space Rⁿ.Asusual, x·y denotes thestandardinner productofxandy inRⁿ.

Inadditiontoitsdenotingabsolutevalue,withoutconfusionwewilluse|· |todenote the standard Euclidean norm on Rⁿ, often to denote n-dimensional volume, and on occasiontodenotetheabsolutevalueofthedeterminantofann×nmatrix.

Forx∈Rⁿ,letx=|x|⁻¹x,wheneverx= 0.

Throughout,Eⁿ isused exclusivelyto denotetheclassof origin-symmetricellipsoids inRⁿ. Wewrite Koⁿ for theset of convex bodiesinRⁿ thatcontainthe originintheir interiors.Thesupportfunctionofaconvex bodyK∈ Koⁿ,hK, isdefinedforallx∈Rⁿ by hK(x) = max{x·y : y ∈ K}. If T ∈ GL(n), then for the support function of the imageT K={T x:x∈K}, weobviouslyhave

hT K(x) =hK

T^tx

, (2.1)

whereT^tdenotes thetranspose ofT.

The set Kⁿo is often equipped with the Hausdorff metric δ_H, which is defined for K₁,K₂∈ Kⁿo byδ_H(K₁,K₂)= max_Sn−1|h_K1−h_K2|.

The classical Aleksandrov–Fenchel–Jessen surface area measure, S_K, of the convex bodyK canbedefinedas theuniqueBorelmeasure onSⁿ⁻¹ suchthat

Sⁿ⁻¹

f(u)dSK(u) =

∂K

f γK(y)

dHⁿ⁻¹(y)

for each continuous f : Sⁿ⁻¹ → R, where γK(y) is the outer unit normal of ∂K at y ∈ ∂K. Recall that γK exists almost everywhere for y ∈ ∂K with respect to the (n−1)-dimensionalHausdorffmeasure Hⁿ⁻¹on∂K.

The cone-volume measure, VK, of the convex body K is a Borel measure on Sⁿ⁻¹ definedforaBorelsetω⊆Sⁿ⁻¹ by

V_K(ω) = 1 n

ω

h_KdS_K.

It is convenient to use thenormalized cone-volumemeasure VK = ^V_|K|^K,of K. Observe that VK is a probability measure on Sⁿ⁻¹. Also, VK is GL(n)-invariant, i.e., for T ∈ GL(n) andaBorelsubsetω⊆Sⁿ⁻¹,ityields

V_T^t_K(ω) =V_K T ω

, (2.2)

where T ω={T u:u∈ω}.

Theprojectionbody,ΠK,oftheconvexbodyK∈ Kⁿo,isaconvexbodyinKoⁿwhose supportfunctionisdefinedforu∈Sⁿ⁻¹ by

h_ΠK(u) =1 2

Sⁿ⁻¹

|u·v|dS_K(v).

According to thedefinitionsofΠK,S_K and V_K, itimmediately yields thatforu∈ Sⁿ⁻¹,

2hΠK(u) n|K| =

Sⁿ⁻¹

|u·v|

h_K(v)dV_K(v).

A finitepositiveBorelmeasure μonSⁿ⁻¹issaidto beisotropicif

Sⁿ⁻¹

(u·v)²dμ(u) =|μ| n ,

forallv∈Sⁿ⁻¹.Here,|μ|denotesthetotalmassofμ.Fornonzerox∈Rⁿ,thenotation x⊗xrepresentstherank1linearoperatoronRⁿthattakesy to(x·y)x.Itimmediately givesthat

tr(x⊗x) =|x|². Equivalently,μisisotropicif

Sⁿ⁻¹

u⊗udμ(u) = |μ| n I_n,

where I_n denotes theidentity operatoron Rⁿ. For moreinformation aboutthe impor- tance ofisotropy,onecanreferto[6,16,17,41,44].

LetΦbetheclassofconvexfunctionsϕ: [0,∞)→[0,∞) thatarestrictlyincreasing and satisfy ϕ(0)= 0. We saythata sequence {ϕ_i}i∈N ⊂Φis such thatϕ_i → ϕ₀ ∈Φ, provided

|ϕi−ϕ0|I := max

t∈I

ϕi(t)−ϕ0(t) →0, foreachcompactintervalI⊂[0,∞).

LetμbeafiniteBorelmeasureonSⁿ⁻¹.Foracontinuousfunctionf :Sⁿ⁻¹→[0,∞), theOrlicznormfϕ off isdefinedby

fϕ= inf

λ >0 : 1

|μ|

Sⁿ⁻¹

ϕ f

λ

dμ≤ϕ(1)

.

Thefollowing Lemma 2.1was previouslyprovedin[22].Lemma 2.2 isexplicitly ap- pearedintheproofofLemma 2.1,whichwillbeusedfrequentlythroughoutthis paper.

Lemma2.1. Supposeμ isa finite Borelmeasureon Sⁿ⁻¹ and thefunction f :Sⁿ⁻¹ → [0,∞)iscontinuousandsuchthatμ({f = 0})>0.ThentheOrlicznormf_ϕispositive and

fϕ=λ0 ⇐⇒ 1

|μ|

Sⁿ⁻¹

ϕ f

λ0

dμ=ϕ(1).

Lemma2.2. Supposeμ isa finite Borelmeasureon Sⁿ⁻¹ and thefunction f :Sⁿ⁻¹ → [0,∞)is continuousandsuchthat μ({f = 0})>0.Thenthefunction

ψ(λ) :=

Sⁿ⁻¹

ϕ f

λ

dμ, λ∈(0,∞),

has thefollowingproperties:

(1) ψ iscontinuous andstrictlydecreasing in(0,∞);

(2) limλ→0⁺ψ(λ)=∞; (3) limλ→∞ψ(λ)= 0;

(4) 0< ψ⁻¹(a)<∞foreach a∈(0,∞).

ForK,L∈ Koⁿ andε≥0,wedefinethefunctionh_ϕ,ε:Rⁿ →[0,∞) as h_ϕ,ε(x) = inf

λ >0 :ϕ

hK(x) λ

+εϕ

hL(x) λ

≤ϕ(1)

.

Observe that h_ϕ,ε is both sublinear and positive when x = 0. Hence, there exists a uniqueconvexbodyK+_ϕ,εL∈ Kⁿo such thatwhose supportfunctionis preciselyh_ϕ,ε. AccordingtoLemmas 8.2and8.4in[15],itgivesthat

K+_ϕ,εL→K, as ε→0⁺, and

ε→0lim⁺

h_K+ϕ,εL(u)−h_K(u)

ε = h_K(u) ϕ₋(1)ϕ

h_L(u) hK(u)

, (2.3)

uniformly for u ∈ Sⁿ⁻¹, where ϕ₋(1) denotes the left derivative of ϕ at 1.Note that h(ε,u)=hϕ,ε(u): [0,∞)×Sⁿ⁻¹→[0,∞) isjointlycontinuousinεandu(seeLemma A.1 in Appendix A for details). So, by (2.3) and Aleksandrov’s variational principle (see Lemma3.1in[8]orLemma8.3 in[15]),ityields that

ε→0lim⁺

|K+_ϕ,εL| − |K|

ε =

Sⁿ⁻¹ ε→0lim⁺

h_K+ϕ,εL(u)−h_K(u) ε dS_K(u)

= n

ϕ₋(1)

Sⁿ⁻¹

ϕ hL

h_K

dVK.

Now, weareinthepositionto givethedefinitionofOrliczmixedvolume.

Definition 2.3.LetK,L∈ Koⁿ andϕ∈Φ.Thegeometric quantity Vϕ(K, L) =

Sⁿ⁻¹

ϕ h_L

hK

dVK

iscalled theOrlicz mixedvolumeofK andLregardingϕ.ThenormalizedOrlicz mixed volume Vϕ(K,L),ofK andLregardingϕ,isdefinedby

Vϕ(K, L) =ϕ⁻¹

Vϕ(K, L)

|K|

=ϕ⁻¹

Sⁿ⁻¹

ϕ hL

hK

dVK

.

Ifϕ(t)=t^p,1≤p<∞,thenVϕ(K,L) andVϕ(K,L) reducetotheLp mixedvolume Vp(K,L) andthenormalizedLp mixedvolumeVp(K,L) used in[38], respectively.

Lemma 2.4.Suppose K,L∈ Kⁿo andϕ∈Φ.Then (1) Vϕ(K,K)= 1.

(2) Vϕ(K,K)=ϕ(1)|K|.

(3) Vϕ(K,λL)=Vϕ(λ⁻¹K,L),forallλ>0.

(4) V_ϕ(K,T L)=V_ϕ(T⁻¹K,L),forallT ∈GL(n).

(5) Vϕ(K,T L)=|T|Vϕ(T⁻¹K,L),forallT ∈GL(n).

Proof. FromDefinition 2.3,itimmediately gives(1) and(2).

Combining Definition 2.3withthefactsthat hλL

hK

= hL

h_λ−1K

and VK=V_λ−1K, ityields (3)directly.

From(2.1),(2.2)andDefinition 2.3,itfollows that Vϕ(K, T L) =ϕ⁻¹

Sⁿ⁻¹

ϕ

hL(T^tu) h_K(T^−tT^tu)

dVK(u)

=ϕ⁻¹

Sⁿ⁻¹

ϕ

h_L(T^tu) h_T−1K(T^tu)

dV_T−1K

T^tu

=V_ϕ

T⁻¹K, L .

FromDefinition 2.3and(4), itfollowsthat ϕ⁻¹

Vϕ(K, T L)

|K|

=ϕ⁻¹

Vϕ(T⁻¹K, L)

|T⁻¹K|

,

whichyields(5)directly. 2

FollowingGHW [15],weintroducetheimportantgeometricquantityV_ϕ(K,L).

Definition2.5.LetK,L∈ Kⁿo andϕ∈Φ.Thegeometricquantity Vϕ(K, L) = inf

λ >0 :

Sⁿ⁻¹

ϕ hL

λh_K

dVK ≤ϕ(1)

iscalledthequasi-Orlicz mixedvolumeofK andLregardingϕ.

If ϕ(t) = t^p, 1 ≤ p < ∞, then Vϕ(K,L) also reduces to the normalized Lp mixed volumeV_p(K,L).Since

Vϕ(K, L) = hL

h_K

ϕ

,

where· ϕistheOrlicznormwithrespecttothemeasure VK,weobtainthefollowing lemmaimmediately.

Lemma2.6. SupposeK,L∈ Kⁿo andϕ∈Φ.Then (1) Vϕ(K,K)= 1.

(2) Vϕ(K,λL)=Vϕ(λ⁻¹K,L)=λVϕ(K,L),forallλ>0.

(3) V_ϕ(T K,L)=V_ϕ(K,T⁻¹L),forallT ∈GL(n).

Inwhatfollows,we provideasimpleidentity, whichwill beusedfrequently.

Lemma 2.7.Suppose K,L∈ Kⁿo andϕ∈Φ.Then V_ϕ

K, L V_ϕ(K, L)

= 1.

Proof. FromDefinition 2.3,Definition 2.5,and thenLemma 2.1,itfollowsthat ϕ

Vϕ

K,Vϕ(K, L)⁻¹L

=

Sⁿ⁻¹

ϕ

h_L Vϕ(K, L)hK

dVK =ϕ(1).

Thus,

V_ϕ

K, L V_ϕ(K, L)

= 1, as desired. 2

3. ThecontinuityofOrlicz mixedvolumes

Inthissection,weshowthecontinuityofthefunctionalsVϕ(K,L) andVϕ(K,L) with respect toϕ,K andL, whichwill beneededinSection5.

Theorem 3.1. Suppose K,K_i,L,L_j ∈ Kⁿo and ϕ,ϕ_k ∈Φ,where i,j,k∈N. If K_i →K, L_j→L andϕ_k→ϕ,then

i,j,klim→∞V_ϕk(K_i, L_j) =V_ϕ(K, L), and

i,j,k→∞lim Vϕk(Ki, Lj) =Vϕ(K, L).

Proof. Let

cm= inf({min_Sn−1hL} ∪ {min_Sn−1hLj :j∈N}) sup({max_Sⁿ−1h_K} ∪ {max_Sⁿ−1h_Ki :i∈N}), and

cM = sup({max_Sn−1h_L} ∪ {max_Sn−1h_Lj :j∈N}) inf({min_Sn−1hK} ∪ {min_Sn−1hKi :i∈N}) . First,weprove

0< cm≤cM <∞.

From the definition of Hausdorff metric, we know that Ki → K and Lj → L are equivalent to hKi →hK and hLj →hL uniformly on Sⁿ⁻¹, respectively. Furthermore, sincehK, hL, hKi,and hLj,i,j∈N,are strictlypositive onSⁿ⁻¹, itfollowsthatthere existsanN₀∈N,suchthatforalli,j > N₀ andu∈Sⁿ⁻¹,

min

Sⁿ⁻¹h^K

2 ≤hKi(u)≤max

Sⁿ⁻¹h2K and min

Sⁿ⁻¹h^L

2 ≤hLj(u)≤max

Sⁿ⁻¹h2L. Let

bm= min

Sminⁿ⁻¹h^K

2,min

Sⁿ⁻¹h^L

2, min

1≤i≤N0

Sminⁿ⁻¹hKi, min

1≤j≤N0

Sminⁿ⁻¹hLj

,

and

b_M = max max

Sⁿ⁻¹h_2K,max

Sⁿ⁻¹h_2L, max

1≤i≤N0

max

Sⁿ⁻¹h_Ki, max

1≤j≤N0

max

Sⁿ⁻¹h_Lj .

Then,0< bm< bM <∞.Meanwhile, wehave

b_mB⊆K⊆b_MB, b_mB⊆K_i⊆b_MB, fori∈N, bmB ⊆L⊆bMB, bmB⊆Lj ⊆bMB, forj∈N. Thus,bythedefinitionsofc_mandc_M,ityields

0< b_m

bM ≤cm≤cM ≤b_M bm

<∞, asdesired.

Next,weprove

i,j,k→∞lim Vϕk(Ki, Lj) =Vϕ(K, L).

Letε>0.Three observationsareinorder.

Firstly,since{ϕ_k}convergesuniformlytoϕon[c_m,c_M],byc_m≤ ^h_h^Lj_Ki^(u)_(u) ≤c_M forall u∈Sⁿ⁻¹,there existsanN₁∈N,suchthatforallk≥N₁,

ϕ_k

h_Lj(u) hKi(u)

−ϕ

h_Lj(u) hKi(u)

< ε

3 (3.1)

holdsindependentlyofiand j anduniformlyforu∈Sⁿ⁻¹. Secondly,thereexistsanN₂∈N,suchthat

ϕ

h_Lj(u) hKi(u)

−ϕ

h_L(u) hK(u)

< ε

3 (3.2)

holds uniformly foru∈Sⁿ⁻¹ and foralli,j≥N2.Indeed,sincetheconvex functionϕ is Lipschitzianon[cm,cM],there existsaconstantC >0,suchthatforallu∈Sⁿ⁻¹,

ϕ

hLj(u) h_Ki(u)

−ϕ

hL(u) h_K(u)

≤C hLj(u)

h_Ki(u)− hL(u) h_K(u)

≤C· δ_H(L_j, L) max_Sn−1h_K+δ_H(K_i, K) max_Sn−1h_L min_Sn−1hKi·min_Sn−1hK

.

Thirdly,sincethe measuresequence {V_Ki}weaklyconvergesto V_K,there existsan N₃∈N, suchthatforalli≥N₃,

Sⁿ⁻¹

ϕ hL

h_K

dV_Ki−

Sⁿ⁻¹

ϕ hL

h_K

dV_K < ε

3. (3.3)

Intermsof(3.1),(3.2)and(3.3),itfollowsthatforalli,j,k≥max{N1,N2,N3},

Sⁿ⁻¹

ϕ_k h_Lj

hKi

dV_Ki−

Sⁿ⁻¹

ϕ h_L

hK

dV_K

≤

Sⁿ⁻¹

ϕk

hLj

h_Ki

−ϕ hLj

h_Ki

dVKi

+

Sⁿ⁻¹

ϕ h_Lj

hKi

−ϕ h_L

hK

dV_Ki +

Sⁿ⁻¹

ϕ hL

h_K

dVKi−

Sⁿ⁻¹

ϕ hL

h_K

dVK

< ε.

Thatis,

i,j,klim→∞

V_ϕk(K_i, L_j)

|Ki| =V_ϕ(K, L)

|K| . Combined withthefact|K_i|→ |K|,ityieldsthefirstconclusion.

Finally,weproceed toprove

i,j,klim→∞V_ϕk(K_i, L_j) =V_ϕ(K, L).

Let

am= inf φ(cm)

∪

φk(cm) :k∈N ,

and

a_M = sup φ(c_M)

∪

φ_k(c_M) :k∈N .

Then,0< a_m≤a_M <∞. Forbrevity,let

ai,j,k= Vϕk(Ki, Lj)

|Ki| and a= Vϕ(K, L)

|K| .

Thus,toshow thedesiredlimit,itsufficesto show

i,j,klim→∞ϕ_k⁻¹(a_i,j,k) =ϕ⁻¹(a).

Since

ϕ_l→ϕ ⇒ ϕ_l→ϕ, uniformly on [c_m, c_M], itimplies

ϕ_l⁻¹→ϕ⁻¹, uniformly on [a_m, a_M].

Notethat

a_i,j,k, a∈[a_m, a_M], for eachi, j, k.

So,foranyε>0,thereexists anN₄∈N,suchthatforalll≥N₄, ϕ_l⁻¹(a_i,j,k)−ϕ⁻¹(a_i,j,k) < ε

2 (3.4)

holds independentlyof i,j and k. Bythefirst conclusionandthe continuityof ϕ⁻¹ on [am,aM], thereexistsanN5∈N,suchthatforalli,j,k > N5,

ϕ⁻¹(ai,j,k)−ϕ⁻¹(a) <ε

2. (3.5)

Intermsof(3.4)and(3.5),itimpliesthatfori,j,k≥max{N4,N5}, ϕ_k⁻¹(a_i,j,k)−ϕ⁻¹(a) < ε.

Thiscompletestheproof. 2

Theorem 3.2. Suppose K,Ki,L,Lj ∈ Kⁿo and ϕ,ϕk ∈Φ,where i,j,k∈N. If Ki →K, Lj→L andϕk→ϕ,then

i,j,klim→∞V_ϕk(K_i, L_j) =V_ϕ(K, L).

Proof. Letc_m andc_M bethenumbersintheproofofTheorem 3.1.Foreachi,j,k, let ψ(i,j,k)(λ) =

Sⁿ⁻¹

ϕk

h_Lj λhKi

dV_Ki,

ψ(λ) =

Sⁿ⁻¹

ϕ hL

λh_K

dVK,

ψ^(k)_m (λ) =ϕ_k cm

λ

,

ψ^(k)_M (λ) =ϕk

cM

λ

,

where λ∈(0,∞).

Forbrevity, let

λ₀=V_ϕ(K, L) and λ_(i,j,k)=V_ϕk(K_i, L_j).

From

ψ^(k)_m ≤ψ≤ψ^(k)_M , ψ^(k)_m ≤ψ_(i,j,k)≤ψ^(k)_M , fori, j, k∈N, and Lemma 2.2, ityieldsthat

c_m=

ψ_m^(k)₋1 ϕ_k(1)

≤(ψ_(i,j,k))⁻¹ ϕ_k(1)

≤

ψ_M^(k)₋1 ϕ_k(1)

=c_M. Thatis,

cm≤λ(i,j,k)≤cM, fori, j, k∈N.

Thus, to show that the sequence {λ_(i,j,k)}i,j,k converges to λ0 as i,j,k → ∞, it suf- fices to show each convergent subsequence {λ_(ip,jq,kr)}p,q,r∈N converges to λ0 when i_p,j_q,k_r→ ∞.

Assumethatlim_p,q,r→∞λ_(ip,jq,kr)=λ₀.FromLemma 2.7andTheorem 3.1,itfollows that

1 = lim

p,q,r→∞Vϕ_kr

Kip, Ljq

λ_(ip,jq,kr)

=Vϕ

K, L

λ0

,

whichin turn givesψ(λ0)=ϕ(1). Lemma 2.2 guaranteesthatλ0 =ψ⁻¹(ϕ(1))=λ0. Thiscompletestheproof. 2

4. Orlicz–Johnellipsoids

Inthissection,wefocus onthemain ProblemSϕ posedinSection1.

Problem Sϕ. Given a convex body K in Rⁿ that contains the origin in its interior, find anellipsoid E, amongst all origin-symmetricellipsoids, whichsolves the following constrainedmaximizationproblem:

maxE |E| subject to V_ϕ(K, E)≤1.

Lemma4.1. Thereexistsasolution toProblemSϕ.

Proof. Given an ellipsoid E ∈ Eⁿ, we use d_E to denote its maximal principalradius.

ThereexistsavE∈Sⁿ⁻¹suchthatdE|vE·u|≤hE(u) forallu∈Sⁿ⁻¹.Fromthestrict monotonicityandconvexityofϕ, togetherwithJensen’sinequality,itfollowsthat

2dE

n|K| min

Sⁿ⁻¹hΠK≤ 2d_E

n|K|hΠK(vE)

=

Sⁿ⁻¹

dE|u·vE|

h_K(u) dVK(u)

≤ϕ⁻¹

Sⁿ⁻¹

ϕ

dE|u·vE| hK(u)

dVK(u)

≤ϕ⁻¹

Sⁿ⁻¹

ϕ h_E

hK

dV_K

=V_ϕ(K, E).

LetEϕ={E∈ Eⁿ :Vϕ(K,E)≤1}.Then,theaboveinequalitiesyieldthat dE≤ n|K|

2min_Sⁿ−1h_ΠK,

for all E ∈ Eϕ. Thus, the set Eϕ is bounded in the metric space (Eⁿ,δ_H). According to Theorem 3.1, the functional V_ϕ(K,·) is continuous. So, Eϕ is also closed. By the Blaschkeselectiontheorem,eachmaximizingsequence ofellipsoids forProblemSϕ has aconvergentsubsequencewhoselimitisstillinEϕ.Therefore,asolutiontoProblemSϕ

exists. 2

Theorem4.2. Thereexistsauniquesolution toProblem Sϕ.

Proof. We argue by contradiction. Assume that there are two different solutions E1

and E2 to Problem Sϕ. Let E1 = T1B and E2 = T2B, where T1,T2 ∈ GL(n). Then, det(T1)= det(T2) andϕ(Vϕ(K,Ei))≤ϕ(1),fori= 1,2.

SinceeachT ∈GL(n) canberepresentedintheformT =P Q,whereP issymmetric, positive definite and Q is orthogonal, we may assume that T₁ and T₂ are symmetric and positivedefinite.ThenT₁ =λT₂, forallλ>0.From theMinkowskiinequalityfor positivedefinitematrices, itgivesthat

det T₁

2 +T₂ 2

_n¹

>1

2det (T1)ⁿ¹ +1

2det (T2)¹ⁿ. LetE3= ^T1+T₂ ²B.Thenwehave

|E3|>|E1|=|E2|. (4.1) From (2.1)andthetriangleinequality,ityieldsthatforallu∈Sⁿ⁻¹,

hE3(u) =

T₁^t+T₂^t 2 u

≤|T₁^tu|+|T₂^tu|

2 =hE1(u) +hE2(u)

2 . (4.2)

Now,fromDefinition 2.3,themonotonicityofϕtogetherwith(4.2),andtheconvexity of ϕ, itfollowsthat

ϕ

V_ϕ(K, E₃)

=

Sⁿ⁻¹

ϕ hE3

h_K

dV_K

≤

Sⁿ⁻¹

ϕ

hE1+hE2

2h_K

dVK

≤1 2ϕ

V_ϕ(K, E₁) +1

2ϕ

V_ϕ(K, E₂)

≤ϕ(1).

Thatis,E₃ satisfiestheconstraintV_ϕ(K,E₃)≤1.Then, itwill resultin

|E₃| ≤ |E₁|=|E₂|, whichcontradicts (4.1). 2

Uptonow,weprovedtheexistenceanduniquenessofthesolutiontoProblemSϕ.In lightoftheclose connectionofthefunctionalsV_ϕ(K,E) and V_ϕ(K,E),wecangivean alternative formulationto ProblemS_ϕ.

Lemma 4.3.The followingpropositions hold:

(1) max_{E∈En:Vϕ(K,E)≤1}|E|= max_{E∈En:Vϕ(K,E)=1}|E|. (2) {E∈ Eⁿ :Vϕ(K, E) = 1}={E∈ Eⁿ:Vϕ(K, E) = 1}. (3) max_{E∈En:Vϕ(K,E)=1}|E|= max_{E∈En:Vϕ(K,E)≤1}|E|.

Proof. Toprove(1),itsufficesto proveE₁∈ Eⁿ cannot beasolutionto ProblemS_ϕ if V_ϕ(K,E₁)<1.Indeed,Lemmas 2.2 and2.7 imply

0<Vϕ(K, E1)<1, wheneverV_ϕ(K,E₁)<1.Thus,

V_ϕ(K, E₁)⁻¹E₁ >|E₁|. Ontheotherhand,Lemma 2.7guaranteesthat

V_ϕ(K, E₁)⁻¹E₁∈

E∈ Eⁿ:V_ϕ(K, E)≤1 .

Therefore,(1)holds.

ForanellipsoidE∈ Eⁿ,Lemma 2.7guaranteesthat V_ϕ(K, E) = 1 ⇐⇒ V_ϕ(K, E) = 1, whichimmediatelyyields(2).

LetE₂be anellipsoidinEⁿ,suchthatV_ϕ(K,E₂)<1.Lemma 2.6impliesthat V_ϕ

K,V_ϕ(K, E₂)⁻¹E₂

= 1.

Consequently,

Vϕ(K, E2)⁻¹E2∈

E∈ Eⁿ :Vϕ(K, E)≤1 .

However,

Vϕ(K, E2)⁻¹E2 >|E2|.

Frompropositions(1),(2) andLemma 4.1, ityields(3). 2

Atthisstage,wecanreformulateProblem S_ϕ asthe following:given aconvex body K ∈ Kⁿo, find an ellipsoid E, amongst allorigin-symmetric ellipsoids, which solvesthe constrainedmaximizationproblem:

maxE

|E|

ω_n subject to V_ϕ(K, E)≤1.

Following theroutineofLYZ[38], wecansimilarlyproposeProblemSϕ,which isin somesense dualtoProblemSϕ.

Problem S_ϕ. Given a convex body K ∈ Koⁿ, find an ellipsoid E, amongst all origin- symmetric ellipsoids,whichsolvesthefollowing constrainedminimizationproblem:

minE

Vϕ(K, E) subject to |E| ωn ≥1.

Ifϕ(t)=t^p,1≤p<∞,thenProblemSϕturnsto ProblemSp,whichwasoriginally discussedbyLYZin[38].Inwhatfollows,weshowaninterestingfactthatthesolutions to ProblemS_ϕ andProblemS_ϕ onlydifferbyascalefactor.

Theorem 4.4.SupposeK∈ Kⁿo andϕ∈Φ.

(1) LetE_M betheuniquesolution toProblem S_ϕ,then ω_n

|EM| _n¹

E_M

isasolution toProblem S_ϕ.

(2) If E_m isasolution toProblemS_ϕ,then

V_ϕ(K, E_m)⁻¹E_m isasolution toProblem Sϕ.

Consequently, thereexistsa uniquesolution toProblemSϕ. Proof. (1)ForanyE∈ {E∈ Eⁿ:|E| ≥ω_n}, itobviouslyhas

Vϕ(K, E)⁻¹E ∈

E∈ Eⁿ:Vϕ(K, E)≤1 .

So,

|EM| ≥ |Vϕ(K, E)⁻¹E|.

Accordingto Lemma 4.3(1), wehaveVϕ(K,EM)= 1.Hence, V_ϕ(K, E)≥

|E|

|E_M| _n¹

≥ ωn

|E_M| _n¹

=V_ϕ

K, ωn

|E_M| ¹_n

E_M

.

Addedthat(_|E^ωn

M|)ⁿ¹E_M ∈ {E∈ Eⁿ :|E| ≥ω_n},itimpliesthattheellipsoid(_|E^ωn

M|)ⁿ¹E_M is asolutiontoProblemSϕ.