On the L dual Minkowski problem Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

On the L

dual Minkowski problem

Yong Huang^a,1,Yiming Zhao^b,∗

aInstituteofMathematics,HunanUniversity,Changsha,410082,China

bDepartmentofMathematics,NewYorkUniversity,Brooklyn,NY11201,USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received4June2017

Receivedinrevisedform1April 2018

Accepted12April2018 CommunicatedbytheManaging EditorsofAIM

MSC:

52A40 52A38 35J96 35J20

Keywords:

LpdualMinkowskiproblem LpMinkowskiproblem DualMinkowskiproblem

DegenerateMonge–Ampèreequation Regularity

TheLp dualcurvature measurewas introduced byLutwak, Yang & Zhang in an attempt to unify the Lp Brunn–

Minkowski theory and the dual Brunn–Minkowski theory.

ThecharacterizationproblemforLpdualcurvaturemeasure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowskiproblem, twomajorproblems inmodern convex geometrythatremainopeningeneral.Inthispaper,existence resultsontheLp dualMinkowskiproblemintheweaksense willbeprovided.Moreover,existenceanduniquenessof the solutioninthesmoothcategorywillalsobedemonstrated.

©2018ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:huangyong@hnu.edu.cn(Y. Huang),yiming.zhao@nyu.edu(Y. Zhao).

1 Research of the first authorwas supported by the National Science Fund for Distinguished Young Scholars(No. 11625103)andtheFundamentalResearchFundsfortheCentralUniversities.

https://doi.org/10.1016/j.aim.2018.05.002 0001-8708/©2018ElsevierInc.Allrightsreserved.

1. Introduction

TheclassicalBrunn–Minkowskitheoryfocusedonstudyinggeometricinvariantssuch as quermassintegrals (which include volume, surface area, and mean width) and geo- metricmeasures suchas areameasures andFederer’scurvaturemeasures.Amongthese measures are surface area measure and (Aleksandrov’s) integral curvature, two most studiedmeasuresintheBrunn–Minkowskitheory.TheMinkowskiproblemandtheAlek- sandrov problemcharacterizingthese twomeasures areinfluentialproblems notonlyin geometricanalysis,butalsointhetheoryoffullynonlinearpartialdifferentialequations.

The L_p Brunn–Minkowski theory and the dual Brunn–Minkowski theory are two theories that fundamentally extended the classical Brunn–Minkowski theory. The last twodecadessawtherapiddevelopmentofthesetwotheoriesandtheyarenowbecoming thecenterfocusofmodernconvexgeometry.

The Lp Brunn–Minkowski theory came to life when Lutwak [49,50] introduced Lp

surfaceareameasure.When p= 1,the Lp surfaceareameasure is theclassicalsurface areameasure.Since itsintroduction,thefamilyofL_p surfaceareameasure hasquickly becomethetopicofmanyinfluentialworks.TheL_pMinkowskiproblemistheproblemof prescribing L_p surfacearea measure,which greatly generalizes theclassical Minkowski problem. The family of Lp Minkowski problems contains important singular unsolved casessuchasthelogarithmicMinkowskiproblem(seeBöröczky,Lutwak,Yang&Zhang [10])andthecentro-affineMinkowskiproblem.

ThedualBrunn–MinkowskitheorywasintroducedbyLutwak(seeSchneider[58])in the1970s.ThedualBrunn–Minkowskitheoryhasbeenmosteffectiveinansweringques- tionsrelatedtointersections.OnemajortriumphofthedualBrunn–Minkowskitheoryis tacklingthefamousBusemann–Pettyproblem,seeGardner[21],Gardner,Koldobsky&

Schlumprecht[23], Koldobsky[38–40],Lutwak[48], andZhang[67].Overtheyears,the dualtheoryhasproducednumerousprofoundconceptsandresults.SeeGardner[22] and Schneider[58] foranoverviewofthetheory.Thedualtheorymakesextensiveuseoftech- niquesfromharmonicanalysis.Recently,thedualBrunn–Minkowskitheorytookahuge step forwardwhen Huang,Lutwak, Yang &Zhang[33] discoveredthe familyoffunda- mentalgeometricmeasures—calleddualcurvaturemeasures—inthedualtheory.These measures are dual to Federer’s curvature measures and are expected to playthe same important role asareameasures and curvature measuresinthe Brunn–Minkowskithe- ory.ThedualMinkowskiproblemistheproblemofprescribingdualcurvaturemeasures.

ThedualMinkowskiproblemnotonlycontainscriticalproblemssuchasthelogarithmic Minkowski problems and the Aleksandrov problem (prescribing curvature measure) as specialcases, butalsointroducesintrinsic PDEs—somethinglongmissing—tothedual Brunn–Minkowski theory. The dual Minkowski problem, while still largely open, has been studiedin[8,12,30,33,68,69].

ArecentsurprisingdiscoverybyLutwak,Yang&Zhang[53] revealedthatthereexists a unifying theory thatincludes the classicalBrunn–Minkowski theory, the L_p Brunn–

Minkowski theory, and the dual Brunn–Minkowski theory. The latter two were never

thoughttobeconnected.Inparticular,theyintroducedtheLp dualcurvaturemeasure, whichunifiesboththeaforementionedLpsurfaceareameasuresanddualcurvaturemea- sures.Theproblemofprescribingthisunifying familyofmeasuresiscalled theLpdual Minkowskiproblem.

Problem 1.1(The L_p dual Minkowskiproblems).Given a nonzerofinite Borelmeasure μontheunitsphere Sⁿ⁻¹ and realnumbersp,q,whatare thenecessaryand sufficient conditionssothatμisexactlyequaltoCp,q(K,·) forsomeconvex bodyK∈Kⁿo?

TheL_pdualMinkowskiproblemswillbethoroughlyinvestigatedinthecurrentwork.

Whenthegivenmeasureμhasadensityf,theL_p dualMinkowskiproblemsbecomes thefollowingMonge–AmpèretypeequationonSⁿ⁻¹:

det(hij(v) +h(v)δij) =f(v)h^p−1(v)(h²(v) +|∇h(v)|²)^n−q2 , (1.1) where f is a given positive smooth function on Sⁿ⁻¹, h is the unknown, δ_ij is the Kronecker delta, and ∇hand (h_ij) are the gradient and the Hessian of hon the unit spherewithrespectto anorthonormalbasisrespectively.

It is worth noting that the L_p dual Minkowski problem contains the classical Minkowskiproblem and the Aleksandrov problem,as well as theunsolved logarithmic Minkowskiproblem[10,70] andtheunsolvedcentro-affineMinkowskiproblem[19,71].It notonlyunifiestheLp Minkowskiproblem[19,35,49] andthedualMinkowskiproblem posedin[33],butalsoincludestheLpAleksandrovproblem[34] andmanynewproblems.

As mentioned previously, the family of Lp dual curvature measures, or sometimes simply referred to as the (p,q)-th dual curvature measure and denoted by Cp,q(K,·) for q ∈ R, were recently introduced by Lutwak, Yang & Zhang [53] in an effort to continue their groundbreaking works [33,34] with the first named author and to unify conceptsneverthoughttobeconnectedintheLpBrunn–Minkowskitheoryandthedual Minkowskitheory.

Whenq=n,the(p,n)-thdualcurvaturemeasureisuptoafactorofnequaltotheLp

surfaceareameasurewhichisthecoreconceptintheLpBrunn–Minkowskitheory.Inthis case,theL_pdualMinkowskiproblembecomestheL_pMinkowskiproblem,whichincludes theclassical Minkowskiproblem solved byMinkowski, Fenchel& Jessen, Aleksandrov, etc.RegularityresultsontheclassicalMinkowskiproblemincludetheinfluential paper [17] byCheng&Yau.Whenp>1,theL_pMinkowskiproblemwassolvedbyLutwak[49]

andLutwak&Oliker[51] wheneverthegivendataisevenandbyChou&Wang[19] in thegeneralcase.TheL_pMinkowskiproblemwhenp<1 ismuchmorecomplicatedand containslongopen problems suchas thelogarithmicMinkowskiproblem (solvedinthe symmetriccasebyBöröczky,Lutwak,Yang&Zhang[10] withrecentmajor progressin thenon-symmetriccasemadebyChen,Li&Zhu [16])andthecentro-affineMinkowski problem.

The logarithmic Minkowski problem characterizes cone volume measure which has been the central topicina number of recentworks. When thegiven data is even, the

existence of solutionsto the logarithmic Minkowski problem was completely solved in Böröczky, Lutwak,Yang,&Zhang [10].In thegeneralcase(non-even case),important contributions weremade byZhu [70], laterbyBöröczky, Hegedűs&Zhu [6], andespe- ciallybyChen,Li&Zhu[16] morerecently.

Thecentro-affineMinkowskiproblemcharacterizesthecentro-affinesurfaceareamea- surewhosedensityinthesmoothcaseisthecentro-affineGausscurvature. Thecharac- terizationproblem,inthiscase,isthecentro-affineMinkowskiproblemposedinChou&

Wang[19].SeealsoJian,Lu&Zhu[37],Lu&Wang[41],Zhu[71],etc.,onthisproblem.

We emphasize again that the Lp dual Minkowski problem, considered in the cur- rent work,containboth theunsolvedlogarithmicMinkowskiproblemand theunsolved centro-affineMinkowskiproblem.

Whenq= 0,theLp dualMinkowskiproblemsbecomestheLp Aleksandrovproblem posed byHuang,Lutawak,Yang,&Zhang[34],whichistheLp versionoftheAleksan- drov problem.Solutionsinsomespecialcasesweregivenin[34].

When p= 0,theL0 dual Minkowskiproblem becomestheunsolved dual Minkowski problem posedbyHuang, Lutawak,Yang,&Zhang[33]. Whenq <0,acompletesolu- tionto thedual Minkowskiproblem—including theexistenceandtheuniquenessofthe solution—waspresentedin[68]. Whenq= 0,thedual Minkowskiproblem becomesthe AlesksandrovproblemsolvedbyAleksandrovhimselfusingatopologicalargument.The dual Minkowskiproblem getsmuchmorechallenging whenq >0 and onlysolutionsin thecasewhenthegivendataisevenexist.When0< q < n,amasssubspaceinequality wasgiven in[33] and proventobe sufficient.Although theconditionisapparentlynec- essarywhen0< q≤1,examplesofconvexbodiesthatviolatesthegivenmasssubspace inequalitywhen1< q < nwereindependentlydiscoveredbyBör¨roczky,Henk,&Pollehn [8] andZhao[69].A bettersubspacemassinequalitywasproposed in[8,69], whichwas shown to be sufficient for the dual Minkowski problem when q ∈ (1,n) is an integer in [69] and necessary when q ∈ (1,n) (integers or not)in [8]. Very recently,Böröczky, Lutwak,Yang,Zhang&Zhao[12] showedthesufficiencyofthesubspacemassinequality when q∈(1,n) is anon-integer,thussettling theexistencepartofthedual Minkowski problem when q ∈(0,n) andthegiven data iseven. When q=n,the dual Minkowski problembecomesthelogarithmicMinkowskiproblemmentionedabove.Whenq≥n+ 1, Henk&Pollehn[30] recentlyproposedanewsubspacemassinequalityandproveditto be necessaryfortheexistencepartofthedualMinkowskiproblemwhenthegivendata is even. Note again that, except for q = n and q ≤ 0, all existingresults on the dual Minkowskiproblem arerestrictedto thecasewhen thegiven measureiseven.

Evenmorechallengingthanfindingthecorrectnecessaryandsufficientconditionsfor the existenceofsolutionto thosenewly posedMinkowski-type problemsis establishing theuniquenessofthesolution.Sofar,uniquenessofthesolutionhasonlybeenestablished for very few cases including the L_p Minkowski problem when p ≥ 1,the Aleksandrov problem, the dual Minkowski problem when q < 0, and the logarithmic Minkowski problem whenthedimensionis2and thegiven dataiseven(see[9]).

Ascanbeseenfromtheabovediscussion,evenspecialcasesoftheLpdualMinkowski problemscanbe extremely challenging.Inthecurrent work,we willstudy theLp dual Minkowskiproblems inthecasewhenp=q.

In Section 3, we will consider the Lp dual Minkowski problems (when p = q) in theweaksense usingvariational method.Because the behaviorof the functional to be optimizedchangesconsiderablybasedonthevaluesofpandqwithrespectto0,results obtainedherehavetobesplit intothree parts.

Whenp>0 and q <0,acomplete characterizationto existencepartoftheL_p dual Minkowskiproblem willbe given.Thisisanextensionto theexistenceresultsobtained in[68].

Theorem1.2.Letp>0,q <0,andμbeanon-zerofiniteBorelmeasureonSⁿ⁻¹.There existsaconvexbodyK∈Kⁿo suchthatμ=Cp,q(K,·)ifandonlyifμisnotconcentrated onany closed hemisphere.

Whenp,q >0 andp =q, a complete solution to the existencepart of the L_p dual Minkowskiproblem will be presented when thegiven data is even. Setting q= n, the followingtheorem containsthesolutionto theevenLp Minkowskiproblem whenp>0 andp=nwhichwasobtainedinLutwak[49] andHaberl,Lutwak,Yang &Zhang[27].

Theorem 1.3. Let p,q > 0, p= q, and μ be a non-zero even Borel measure on Sⁿ⁻¹. There existsan origin symmetricconvex body K ∈ Kⁿe such that μ=C_p,q(K,·) if and onlyif μisnot concentratedinany great subsphere.

TheLp Minkowskiproblemwhenp≥1 andp=nwassolvedinthegeneralcase(not assumingevennessofthemeasure)inChou&Wang[19] andlaterinHug,Lutwak,Yang

&Zhang[35].Unfortunately,theirsolutionsdonotextendtothiscaseinfullgenerality.

ThekeyobstacleisthelackofaMinkowskitypeinequalityinthismoregeneralsetting.

Inparticular,notethattheLp Minkowskiproblemwhen0< p<1 was studiedbyZhu [72] andwasessentiallysolvedbyChen,Li&Zhu [15] recently.

Whenp,q <0 andp=q, thefollowing resultwillnowbe established.

Theorem 1.4. Let p,q < 0, p = q, and μ be a non-zero finite even Borel measure on Sⁿ⁻¹.Ifμ vanishesonallgreatsubsphere,thenthereexistsanoriginsymmetricconvex bodyK∈Kⁿe suchthat μ=Cp,q(K,·).

NotethattheoptimizationproblemtobeintroducedinSection3failswhenthegiven measureμhaspositiveconcentrationinanygreat subsphere.

Tocomplementandenrichtheresultsobtainedviavariationalmethod,solutionsvia continuitymethodswillbeprovided.InSection4,boththeexistenceandtheuniqueness of thesolution to theL_p dual Minkowskiproblem inthe smooth category will be pre- sented.Note thatwhen p= 0, the following theorem provides regularity resultsto the dualMinkowskiproblemfornegativeindices.

Theorem 1.5. Suppose p > q and 0 < α ≤ 1. For any given positive function f ∈ C^α(Sⁿ⁻¹),there exists aunique solution h∈C^2,α(Sⁿ⁻¹) to(1.1).If f issmooth, then thesolution isalso smooth.

Since whenq=n,thedualMinkowskiproblem becomestheL_p Minkowskiproblem, Theorem1.5alsosolvestheLpMinkowskiproblemforp> nwhichwasoriginallysolved byChou&Wang[19].

The uniqueness part of the Lp dual Minkowski problem when p> q and the given measure is discrete (inthe polytopalcase)wasproved byLutwak, Yang& Zhang[53].

The uniqueness established in Theorem 1.5 settles the case when the given measure possesses asufficientlysmoothdensityandthe solutionisassumedto possesssufficient regularity.Itremainsunknownwhethertheuniquenessresultstillholdsiftheregularity assumption oftheconvexbodyisremoved(see,forexample[57]).

Note again that the uniqueness of the solution to the Lp dual Minkowski problem remainsoneofthebiggestchallengesinMinkowskitypeproblems.Whiletheuniqueness has been established for p ≥ 1 by using the L_p Minkowski inequality [35], very little progress has been made for p < 1. Important recent results in this direction include Andrews [2],Böröczky,Lutwak, Yang&Zhang[9],Choi&Daskalopoulos [18],Huang, Liu&Xu[31],andJian,Lu&Wang[36].

A major accomplishmentin[53] is showing thattheL_p dualcurvature measuresare valuations.See,forexample,Alesker [1], Haberl[26], Haberl&Parapatits[28],Ludwig [42,43,45,46], Ludwig& Reitzner [47], Schuster [59,60], Schuster & Wannerer [61] and thereferencesthereinforimportantvaluationsinthetheoryofconvexbodies.

Acknowledgments. The authors are grateful to the anonymous referee for his/her valuableinput.

2. Preliminaries

This section is divided into three subsections. In the first subsection, basics in the theoryofconvexbodieswillbecovered.Inthesecond subsection,thenotionofLp dual curvaturemeasure,orthe(p,q)-thdualcurvaturemeasure,willbeintroduced.Lastbut notleast,inthethirdsubsection,wewillpresenttheL_p dualMinkowskiproblems—the characterizationproblem for(p,q)-thdualcurvaturemeasure.

2.1. Basicsinthetheory ofconvex bodies

The book[58] by Schneider offersacomprehensiveoverview ofthetheory ofconvex bodies.

Let Rⁿ be then-dimensional Euclidean space. Theunit spherein Rⁿ is denoted by Sⁿ⁻¹. Wewill write C(Sⁿ⁻¹) for thespace of continuousfunctions onSⁿ⁻¹. The sub- set C⁺(Sⁿ⁻¹) of C(Sⁿ⁻¹) containonlypositivefunctionswhereasthesubsetC_e(Sⁿ⁻¹) containonlyeven functions.Wewill alsowriteC_e⁺(Sⁿ⁻¹) forC⁺(Sⁿ⁻¹)∩Ce(Sⁿ⁻¹).

AconvexbodyinRⁿ isacompactconvexsetwithnonemptyinterior.Theboundary ofK iswrittenas ∂K.Denote byKⁿ0 theclassofconvexbodiesthatcontaintheorigin intheirinteriorsinRⁿ andbyKⁿe theclassoforigin-symmetricconvexbodiesinRⁿ.

LetK be acompactconvex subset ofRⁿ. Thesupportfunction hK of K is defined by

h_K(y) = max{x·y:x∈K}, y∈Rⁿ.

Thesupportfunctionh_K isacontinuousfunctionhomogeneousofdegree1.SupposeK containstheorigininitsinterior. TheradialfunctionρK isdefinedby

ρ_K(x) = max{λ:λx∈K}, x∈Rⁿ\ {0}.

The radial function ρK is a continuous function homogeneous of degree −1. It is not hardtoseethatρK(u)u∈∂K forallu∈Sⁿ⁻¹.

Foreachf ∈C⁺(Sⁿ⁻¹),theWulffshape[f] generatedbyf istheconvexbodydefined by

[f] ={x∈Rⁿ :x·v≤f(v), for allv∈Sⁿ⁻¹}. Itisapparentthath_[f]≤f and[h_K]=KforeachK∈Kⁿ0.

The L_p combination of two convex bodies K,L ∈ Kⁿ0 was first studied by Firey and was the starting point of the now rich Lp Brunn–Minkowskitheory developed by Lutwak [49,50]. For t > 0,the Lp combination of K and L, denoted byK+pt·L, is definedtobetheWulff shapegeneratedbyhtwhere

ht=

(h^p_K+th^p_L)^1p, ifp= 0, hKh^t_L, ifp= 0.

Whenp≥1,bytheconvexityof_p norm, wegetthat h^p_K+pt·L=h^p_K+th^p_L.

SupposeK_i isasequenceofconvexbodiesinRⁿ.WesayK_i convergestoacompact convexsubsetK⊂Rⁿ if

max{|h_Ki(v)−h_K(v)|:v∈Sⁿ⁻¹} →0, (2.1) asi→ ∞.IfK containstheorigininitsinterior, equation(2.1) implies

max{|ρKi(u)−ρK(u)|:u∈Sⁿ⁻¹} →0, asi→ ∞.

ForaconvexbodyK∈Kⁿ0,thepolarbodyofK isgivenby K^∗={y∈Rⁿ :y·x≤1, for allx∈K}. It issimpletocheckthatK^∗∈Kⁿ0 and that

hK^∗(x) = 1/ρK(x),

ρK^∗(x) = 1/hK(x), (2.2)

forx∈Rⁿ\ {o}.Moreover,wehave(K^∗)^∗=K.

Bythe definitionofpolar bodyand therelations(2.2),for K,K_i ∈Kⁿ0,we haveK_i converges toK ifandonlyifK_i^∗ convergesto K^∗.

For a compact convex subset K in Rⁿ and v ∈ Sⁿ⁻¹, the supporting hyperplane H(K,v) ofK at visgiven by

H(K, v) ={x∈K:x·v=hK(v)}.

By its definition, the supporting hyperplane H(K,v) is non-empty and contains only boundary points of K. For x ∈ H(K,v), we say v is an outer unit normal of K at x∈∂K.

Letω⊂Sⁿ⁻¹ beaBorelset.TheradialGauss imageofK atω,denotedbyαK(ω), is definedto bethesetofallunitvectorsvsuchthatv isanouter unitnormalofK at someboundarypointuρK(u) whereu∈ω,i.e.,

α_K(ω) ={v∈Sⁿ⁻¹:v·uρ_K(u) =h_K(v) for someu∈ω}.

Whenω ={u}isasingleton,weusuallywrite αK(u) insteadof themorecumbersome notation αK({u}). LetωK be thesubsetof Sⁿ⁻¹ suchthatαK(u) containsmorethan oneelementforeachu∈ωK.ByTheorem2.2.5in[58],thesetωKhassphericalLebesgue measure0.TheradialGaussmapofK,denotedbyαK,isthemapdefinedonSⁿ⁻¹\ωK

thattakeseachpointuinitsdomaintotheuniquevectorinα_K(u).Henceα_K isdefined almost everywhereonSⁿ⁻¹withrespect tothesphericalLebesguemeasure.

Letη ⊂Sⁿ⁻¹beaBorelset.ThereverseradialGaussimageofK,denotedbyα^∗_K(η), is defined to be the set of all radial directions such that the corresponding boundary points haveat leastoneouterunitnormalinη, i.e.,

α^∗_K(η) ={u∈Sⁿ⁻¹:v·uρK(u) =hK(v) for somev∈η}.

When η ={v}is asingleton,weusually write α^∗_K(v) instead ofthe morecumbersome notationα^∗_K({v}).Letη_KbethesubsetofSⁿ⁻¹suchthatα^∗_K(v) containsmorethanone element foreachv ∈ηK. ByTheorem2.2.11 in[58],the setηK hassphericalLebesgue measure 0. Thereverse radialGauss mapof K,denoted byα^∗_K, isthe mapdefined on Sⁿ⁻¹\ηKthattakeseachpointvinitsdomaintotheuniquevectorinα^∗_K(v).Henceα^∗_K is definedalmost everywhereonSⁿ⁻¹ withrespect tothesphericalLebesguemeasure.

2.2. (p,q)-thdual curvature measures

DualquermassintegralsarethefundamentalgeometricinvariantsinthedualBrunn–

Minkowskitheory.Suppose1≤q≤nisaninteger.The(n−q)-thdualquermassintegral ofK∈Kⁿ0, denotedbyWn−q(K),canbeviewed astheaverageofq-dimensionalinter- sectionareas.Thatis,

Wn−q(K) = ωn

ωq

G(n,q)

H^q(K∩ξq)dξq,

whereH^qistheq-dimensionalHausdorffmeasure andtheintegrationiswithrespect to theHaar measureon theGrassmannianG(n,q) containing allq-dimensionalsubspaces ξq inRⁿ.

The dual quermassintegrals have the following integral representation (see Lutwak [48]):

W_n−q(K) = 1 n

Sⁿ⁻¹

ρ^q_K(u)du,

whichimmediatelyallowsustoextendthedefinitiontoallq∈R.

Theexactgeometricmeasuresthatcanbeviewedasthedifferentialsofdualquermass- integralsremained hidden until the ground-breaking workby Huang,Lutwak, Yang &

Zhang[33].Forq= 0,theq-thdualcurvaturemeasureofK∈Kⁿ0,denotedbyC_q(K,·), canbedefinedastheuniqueBorelmeasureonSⁿ⁻¹thatsatisfiesthefollowingequation foreachL∈Kⁿ0:

d dt

t=0

W_n−q(K+₀t·L) =q

Sⁿ⁻¹

logh_L(v)dC_q(K, v). (2.3)

NotethatthereisacorrespondingversionoftheformulathatallowsustodefineC0(K,·), see[33].Dual curvaturemeasures arethenotionsinthedual Brunn–Minkowskitheory thataredual toFederer’scurvaturemeasures.

The(p,q)-thdual curvature measurewasvery recentlyintroducedbyLutwak,Yang

&Zhang[53].Forp,q= 0,the(p,q)-thdual curvaturemeasureofK∈Kⁿ0,denotedby Cp,q(K,·),isdefinedtobetheuniqueBorelmeasureonSⁿ⁻¹thatsatisfiesthefollowing equationforeachL∈Kⁿ0:

d dt

t=0

W_n−q(K+_pt·L) =q

Sⁿ⁻¹

h^p_L(v)dC_p,q(K, v). (2.4)

Notethatwhenp= 0,the(0,q)-thdualcurvaturemeasureisdefinedasin(2.3) sothat they are exactlythe q-th dual curvature measure.There is acorresponding version of

(2.4) thatallowsustodefineCp,q(K,·) forq= 0.Itshouldalsobepointedoutthatthe (p,q)-th dual curvature measure defined in[53] is slightlymore general than whatwe are usinginthecurrentpaper.

Key propertiesofLpdual curvaturemeasureswereshownin[53].Whenviewedas a functionfromthesetKⁿ0 (equippedwiththeHausdorffmetric)tothespaceofBorelmea- suresonSⁿ⁻¹ (equippedwiththeweaktopology),the(p,q)-thdual curvaturemeasure is continuous.Thatis, foreachf ∈C(Sⁿ⁻¹),

i→∞lim

Sⁿ⁻¹

f(v)dC_p,q(K_i, v) =

Sⁿ⁻¹

f(v)dC_p,q(K₀, v),

given thatK₀,K_i ∈Kⁿ0 and K_i converges to K₀.Moreover, itisalsoavaluation.That is,

Cp,q(K∪L,·) +Cp,q(K∩,·) =Cp,q(K,·) +Cp,q(L,·), given thatK,L∈Kⁿ0 aresuchthatK∪L∈Kⁿ0.

Itwasshownin[53] thatthe(p,q)-thdualcurvaturemeasureisabsolutelycontinuous with respecttotheq-th dualcurvaturemeasure andcanbe writtenas

dC_p,q(K,·) =h⁻_K^pdC_q(K,·).

The above equation works even when p or q is 0. By definition, it is immediate that the (0,q)-th dualcurvature measure is exactlytheq-th dual curvature measure.When q=n,Cn(K,·) isthecone-volumemeasure,

dCn(K,·) = 1

nhKdSK(·), and thusC_p,n(K,·) givestheL_psurfaceareameasure,

dC_p,n(K,·) = 1

nh¹_K^−pdS(K,·) = 1

ndS_p(K,·).

IfKisapolytopethatcontainstheorigininitsinteriorwithouterunitnormalsv_i,i= 1,. . . ,m,thenthe(p,q)-thcurvaturemeasure C˜p,q(K,·) is discreteandis concentrated on{v1,. . . ,vm}.Namely,

C_p,q(K,·) = m i=1

c_iδ_vi(·), where

ci= 1

nhK(vi)^−p

Sⁿ⁻¹∩Δi

ρK(u)^qdu,

andΔ_iis thecone formedbytheoriginandtheface ofK withnormalv_i.

If K is a convex body thathas C² boundary with positive curvature and contains the origin in its interior, then Cp,q(K,·) is absolutely continuous with respect to the Lebesguemeasure,

dCp,q(K, v)

dv = 1

nh^1−p_K (h²_K+|∇hK|²)^q−n2 det((hK)ij+hKδij),

where∇hK and((hK)ij) arethegradientandtheHessianofhKontheunitsphereSⁿ⁻¹ withrespectto anorthonormalbasisrespectively.

2.3. The L_p dual Minkowskiproblems

It is natural to consider the characterization problem for (p,q)-th dual curvature measure.

Problem 2.1(The Lp dual Minkowskiproblems).Given a nonzerofinite Borelmeasure μontheunitsphere Sⁿ⁻¹ and realnumbersp,q,whatare thenecessaryand sufficient conditionssothatthereexistsaconvexbodyK∈Kⁿ0 satisfying

C_p,q(K,·) =μ?

Whenp= 0,theL_pdual MinkowskiproblemsbecomesthedualMinkowskiproblem, which sinceits introduction in[33] has quicklyestablished its fundamentalrole inthe dual Brunn–Minkowski theory already generating works such as [8,12,30,33,68,69]. In particular, thecaseof p= 0,q=nis thelogarithmic Minkowskiproblem while p= 0, q= 0,itistheAleksandrovproblem.

Whenq =n, the Lp dual Minkowskiproblems becomesthe Lp Minkowskiproblem which is fundamentalinthe Lp Brunn–Minkowskitheory. The Lp Minkowskiproblem when p > 1 was solved in the even case by Lutwak [49] and in the general case by Chou–Wang [19]. A different approach was provided by Hug, Lutwak, Yang & Zhang [35].AsmoothsolutionwasobtainedbyHuang&Lu[32] for2< p< n.Thesolutionof theLpMinkowskiproblemplaysavitalroleinestablishingvariouspowerfulsharpaffine isoperimetricinequalities,see, e.g.,Cianchi,Lutwak,Yang&Zhang[20],Lutwak,Yang

&Zhang [52], and Zhang [66]. It containsimportant and unsolved singularcases such asthelogarithmic Minkowskiproblem(p= 0) andthecentroaffineMinkowskiproblem (p=−n).

ThelogarithmicMinkowskiproblem studies conevolumemeasures thatappearedin agrowing numberof works.See,for example,Barthe, Guédon, Mendelson&Naor [4],

Böröczky&Henk[7],Böröczky,Lutwak,Yang&Zhang[9–11],Henk&Linke[29],Lud- wig [44],Ludwig&Reitzner[47],Naor[54],Naor&Romik[55],Paouris&Werner[56], Stancu [62,63],Xiong[65], Zhu[70],andZou&Xiong[73]. ThelogarithmicMinkowski problem has strong connections with isotropic measures (Böröczky, Lutwak, Yang &

Zhang [11]), curvature flows (Andrews [2,3]), and thelog-Brunn–Minkowski inequality (e.g.,Böröczky,Lutwak,Yang&Zhang[9],Xi&Leng[64]),aninequalitystrongerthan theclassicalBrunn–Minkowskiinequality.

Whenthegivenmeasureμhasadensityf,theL_p dualMinkowskiproblemsisequiv- alent tothefollowingMonge–AmpèreequationonSⁿ⁻¹,

det(hij(v) +h(v)δij) =f(v)h^p−1(v)(h²(v) +|∇h(v)|²)^n−q2 , v∈Sⁿ⁻¹. (2.5) Since theunitballsoffinite dimensionalBanachspacesareorigin-symmetricconvex bodiesandthedualcurvaturemeasureofanorigin-symmetricconvexbodyiseven,itis of greatinteresttostudy thefollowing evenLp dualMinkowskiproblems.

Problem2.2(The evenL_p dual Minkowskiproblems).GivenanonzeroevenfiniteBorel measure μon theunitsphere Sⁿ⁻¹ and real numbersp,q, whatare thenecessary and sufficient conditionsso thatthere existsaconvex bodyK∈Kⁿe satisfying

Cp,q(K,·) =μ?

In the current work, we will study the Lp dual Minkowski problems when p = q from two different angles. One is from convex geometric analysis point of viewwhere we will solvethe Lp dual Minkowskiproblems using variationalmethod. The solution will include cases when the given measure possesses no density. This will be done in Section3.Inadditionto that,wewillsolvetheMonge–Ampèreequation(2.5) byusing continuity methods and Caffarelli’s regularity results of the Minkowski problem [13, 14]. Both existence and uniqueness of the solution (in the smooth category) will be demonstrated.Thiswill bedoneinSection4.

3. Avariationalmethodto weaksolutions

Inthissection,wewillusevariationalmethodtostudyweaksolutionstotheLpdual Minkowski problems when p= q and p,q = 0. Note thatwhen p= 0, this problem is known as thedual Minkowskiproblem,and when q= 0, this problem isknown as the L_p Aleksandrovproblem.Boththese specialcaseshavealreadybeenconsidered.

3.1. Anassociatedoptimization problem

To obtain aweak solutionusing variational method, thefirst step is to convert the existenceproblem intoanoptimizationproblem whoseoptimizerisexactlythesolution to theoriginalproblem.

Forany nonzerofinite Borelmeasure μ onSⁿ⁻¹ and p,q= 0, defineΦp,q : Kⁿo →R by

Φ_p,q(Q) =−1 plog

Sⁿ⁻¹

h_Q(v)^pdμ(v) +1 qlog

Sⁿ⁻¹

ρ_Q(u)^qdu,

foreachQ∈Kⁿo.

Weconsiderthemaximizationproblem

sup{Φp,q(Q) :Q∈Kⁿo}.

Lemma 3.1. Letp,q = 0 andμ be a nonzerofinite Borel measureon Sⁿ⁻¹.If K ∈Kⁿo

satisfies

Sⁿ⁻¹

h^p_K(v)dμ(v) =Wn−q(K), (3.1)

and

Φp,q(K) = sup{Φp,q(Q) :Q∈Kⁿo}, (3.2) then

μ=C_p,q(K,·).

Proof. Foreachf ∈C⁺(Sⁿ⁻¹),let[f] betheWulffshapegenerated byf, i.e., [f] ={x∈Rⁿ :x·v≤f(v) for allv∈Sⁿ⁻¹} ∈Kⁿo.

Definethefunctional Ψp,q :C⁺(Sⁿ⁻¹)→Rby Ψp,q(f) =−1

plog

Sⁿ⁻¹

f(v)^pdμ(v) +1 qlog

Sⁿ⁻¹

ρ_[f](u)^qdu.

NotethatΨp,q(f) ishomogeneousofdegree0,i.e.forallλ>0 andf ∈C⁺(Sⁿ⁻¹), Ψp,q(λf) = Ψp,q(f).

Weclaimthat

Ψ_p,q(f)≤Ψ_p,q(h_K), (3.3)

foreachf ∈C⁺(Sⁿ⁻¹).Indeed,sinceh_[f]≤f and[hK]=K, wehave

Ψp,q(f)≤Ψp,q(h[f]) = Φp,q([f])≤Φp,q(K) = Ψp,q(hK), where thesecondinequalitysignfollowsfrom(3.2).

Forany g∈C(Sⁿ⁻¹) andt∈(−δ,δ) where δ >0 issufficiently small,let

ht(v) =hK(v)e^tg(v). (3.4)

ByTheorem4.5 of[33], d

dtW_n−q([h_t]) t=0

=q

Sⁿ⁻¹

g(v)dC_q(K, v). (3.5)

By(3.3),thedefinitionofΨ_p,q,and(3.5),wehave 0 = d

dtΨp,q(ht) t=0

= d dt

⎛

⎝−1 plog

Sⁿ⁻¹

ht(v)^pdμ(v) +1

qlogWn−q([ht])

⎞

⎠

t=0

=−

⎛

⎝

Sⁿ⁻¹

h^p_K(v)dμ(v)

⎞

⎠

−1

Sⁿ⁻¹

h_K(v)^pg(v)dμ(v) + 1 Wn−q(K)

Sⁿ⁻¹

g(v)dC_q(K, v).

By(3.1),wehave

hK(v)^pdμ(v) =dCq(K, v).

Weconcludethatμ=C_p,q(K,·). 2

Wheneverp=q,wecantakeadvantageofthedifferentdegreesofhomogeneityofthe two sidesin(3.1) andgetthefollowing lemma.

Lemma 3.2. Let p,q = 0, p= q, and μ be a nonzero finite Borelmeasure on Sⁿ⁻¹. If K∈Kⁿo satisfies

Φp,q(K) = sup{Φp,q(Q) :Q∈Kⁿo}, then thereexistsaconstant c>0such that

μ=C_p,q(cK,·).

Proof. Notethatsincep=q, wemayfindaconstantc>0 suchthat

Sⁿ⁻¹

h^p_cK(v)dμ(v) =W_n−q(cK).

SinceΦ_p,q ishomogeneousofdegree0,wehave

Φp,q(cK) = sup{Φp,q(Q) :Q∈Kⁿo}. ByLemma3.1,wehave

μ=Cp,q(cK,·). 2

WeremarkthatthereisanobviouswaytoadaptLemmas3.1and3.2andtheirproofs so that they canbe used to treat the case when the given measure μ is even and the solutionconvexbodyKisorigin-symmetric.Inparticular,wehavethefollowinglemma.

Lemma3.3. Letp,q= 0,p=q,andμbe anonzeroevenfinite Borelmeasureon Sⁿ⁻¹. If K∈Kⁿe satisfies

Φp,q(K) = sup{Φp,q(Q) :Q∈Kⁿe}, thenthere existsaconstant c>0suchthat

μ=Cp,q(cK,·).

Lemmas3.2and3.3converttheexistencepartoftheLpdualMinkowskiprobleminto anoptimizationproblem.Therestofthissectionisaimed toproveanoptimizerexists.

3.2. Existence ofan optimizer

Since the behavior of Φ_p,q changesbased on the values of pand q, we shall divide ourresultsinto three cases:i)p>0 and q <0;ii) p,q >0 andp=q; iii) p,q <0 and p=q.Thecasep<0,q >0 remains unsolved.Progressalongthatdirectionisofgreat interest.

Letusfirstdealwiththecasep>0,q <0.

Lemma3.4. Letp>0,q <0,andμbe anon-zerofinite BorelmeasureonSⁿ⁻¹.There exists aconvex body K ∈ Kⁿo such that Φ_p,q(K) = sup{Φ_p,q(Q): Q ∈Kⁿo}if μ is not concentratedonany closed hemisphere.

Proof. SupposeQlis amaximizationsequence;i.e.,

llim→∞Φ_p,q(Q_l) = sup{Φ_p,q(Q) :Q∈Kⁿo}.

Note thatΦp,q is homogeneousofdegree0.Sowe may(after rescaling)assumethat

Sⁿ⁻¹ρ^q_Ql(u)du= 1.

WeclaimthatQ^∗_l isuniformlybounded.Ifnot,aftertakingsubsequence,wemayfind v_l ∈Sⁿ⁻¹ such thatρ_Q^∗_l(v_l)→ ∞ as l → ∞. By thedefinition ofpolar body and the supportfunction,

1 =

Sⁿ⁻¹

ρ^q_Q

l(u)du

=

Sⁿ⁻¹

h^−q_Q∗ l(u)du

≥

Sⁿ⁻¹

ρ_Q^∗_l(v_l)^−q(v_l·u)^−q₊ du

=ρ_Q^∗_l(v_l)^−q

Sⁿ⁻¹

(v₁·u)⁻₊^qdu.

Here (t)₊ = max{t,0} for any t ∈ R. Since

Sⁿ⁻¹(v₁·u)⁻₊^qdu is positive and q < 0, we have ρQ^∗_l(vl) is bounded. This is a contradiction to the choice of vl. Hence Q^∗_l is uniformly bounded.

By Blaschke’s selection theorem, (after taking asubsequence) we may assume that Q^∗_l converges toacompactconvex subsetK0⊂Rⁿ.

We will show thatK0 hasthe origininits interior. Ifnot, then theorigin ison the boundaryof K₀ andtherefore, thereexists u₀∈Sⁿ⁻¹ suchthath_K0(u₀)= 0.Since Q^∗_l converges toK₀,wehave

l→∞lim hQ^∗_l(u0) = 0.

Foreachδ >0,define

ωδ={v∈Sⁿ⁻¹:v·u0> δ}.

Since μ is not concentrated in any closed hemisphere, there exists δ0 > 0 such that μ(ω_δ0)>0.Sinceρ_Q^∗_l(v)v·u₀≤h_Q^∗_l(u₀),wehaveρ_Q^∗_l(v) goesto0 uniformlyonω_δ0.

This, togetherwith

Sⁿ⁻¹ρ^q_Q

l(u)du= 1 andp>0,implies Φp,q(Ql) =−1

plog

Sⁿ⁻¹

h^p_Ql(v)dμ(v)

≤ −1 plog

ωδ0

h^p_Ql(v)dμ(v)

=−1 plog

ωδ0

ρ^−p_Q∗

l(v)dμ(v)

→ −∞,

which isa contradiction to Q_l being amaximizationsequence. HenceK₀ contains the origininitsinterior.LetK =K₀^∗.Since Q^∗_l convergesto K0,Ql convergesto K.That K isamaximizernow followsdirectlyfrom thecontinuityof Φp,q and thefactthatQl

isamaximizing sequence. 2

The “if” part of the following theorem follows directly from Lemmas 3.2 and 3.4 whereasthe“onlyif” partisobvious.

Theorem3.5.Letp>0,q <0andμbeanon-zerofinite BorelmeasureonSⁿ⁻¹.There existsaconvexbodyK∈Kⁿo suchthatμ= ˜C_p,q(K,·)ifandonlyifμisnotconcentrated onany closed hemisphere.

Letusnowconsider thecasep,q >0 andp=q.

Thefollowingtwolemmasprovide importantestimatesforestablishingtheexistence ofanoptimizer.

Lemma3.6.Letp,ε0>0andμbeanon-zeroevenfiniteBorelmeasureonSⁿ⁻¹.Suppose e1l,· · ·,enl isa sequenceof orthonormal basis inRⁿ and {al}isa sequenceof positive real numbers. Assume e_1l,· · ·,e_nl converges to an orthonormal basise₁,· · ·,e_n in Rⁿ. Define

Gl={x∈Rⁿ:|x·e1l|²+· · ·+|x·e_n−1,l|²≤a²_l and |x·en,l|²≤ε0}.

If μ isnot concentrated in any great subsphere, thenthere exists c,L>0 (independent ofl) suchthat

Sⁿ⁻¹

h^p_Gl(v)dμ(v)≥c,

foreach l > L.

Proof. Notethat±ε₀e_n,l∈G_l.Hence,bythedefinitionofsupportfunction,

hGl(v)≥ε0|v·en,l|, (3.6) foreachv∈Sⁿ⁻¹.Foreachδ >0,define

ωδ ={v∈Sⁿ⁻¹:|v·en|> δ}.

Bymonotoneconvergencetheorem,andthefactthatμisnotconcentratedinanygreat subsphere,

δlim→0⁺μ(ωδ) =μ(Sⁿ⁻¹\span{e1,· · ·, en−1})>0 Hence,there existsδ0>0 such thatμ(ωδ0)>0.

Since enl convergestoen,there existsL>0 such thatforl > L,wehave

|enl−en|< δ0/2.

This, togetherwith(3.6),impliesthatforv∈ωδ0 andl > L, h_Gl(v)≥ε₀(|v·e_n| − |v·(e_n−e_nl)|)≥ε₀δ₀/2.

Hence,

Sⁿ⁻¹

h^p_G

l(v)dμ(v)≥

ωδ0

h^p_G

l(v)dμ(v)≥ 1

2ε₀δ₀ p

μ(ω_δ0) =:c. 2

Let0< a<1 ande1,· · ·,en beanorthonormalbasisinRⁿ.Define Ta={x∈Rⁿ :|x·e1| ≤aand |x·e2|²+· · ·+|x·en|²≤1}.

Thefollowinglemma estimatesthedual quermassintegralofthegeneralizedellipsoid T_a.

Lemma 3.7.Let0< a<1ande₁,· · ·e_n bean orthonormalbasis inRⁿ.If q >0,then

alim→0⁺

Sⁿ⁻¹

ρ^q_Ta(u)du= 0.

Proof. Sinceq >0,wehaveρ^q_T

a isbounded.Thus,bydominatedconvergencetheorem,

alim→0⁺

Sⁿ⁻¹

ρ^q_Ta(u)du=

Sⁿ⁻¹

alim→0⁺ρ^q_Ta(u)du.

It issimpletoseethat

alim→0⁺ρTa(u) =

1, ifu∈span{e2,· · ·, en}, 0, otherwise.

Hencewegetthedesiredresult. 2

Weareready toshowtheexistenceofanoptimizer.