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Journal of Functional Analysis
www.elsevier.com/locate/jfa
The L
pcapacitary Minkowski problem for polytopes
✩Ge Xionga,∗, Jiawei Xionga, Lu Xub
aSchoolofMathematicalSciences,TongjiUniversity,Shanghai,200092,China
bCollegeofMathematicsandEconometrics,HunanUniversity,Changsha,410082, China
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received18September2018 Accepted2June2019 Availableonline7June2019 CommunicatedbyGuidoDe Philippis
MSC:
52A20 31B15
Keywords:
LpMinkowskiproblem Capacity
Polytope
Sufficient conditionsaregiven fortheexistence of solutions to the discreteLp Minkowski problem for p-capacity when 0< p<1 and1<p<2.
©2019ElsevierInc.Allrightsreserved.
1. Introduction
Thesetting forthis articleisEuclidean n-space, Rn. A convex body inRn isacom- pactconvexsetwithnonemptyinterior.TheBrunn-Minkowskitheoryofconvexbodies,
✩ ResearchofthefirsttwoauthorsaresupportedbyNSFCNo.11871373;thethirdauthorissupported byNSFCNo.11771132andHunanScienceandTechnologyProjectNo.2018JJ1004.
* Correspondingauthor.
E-mailaddresses:xiongge@tongji.edu.cn(G. Xiong),1610528@tongji.edu.cn(J. Xiong), xulu@hnu.edu.cn(L. Xu).
https://doi.org/10.1016/j.jfa.2019.06.008
0022-1236/©2019ElsevierInc. Allrightsreserved.
also called themixedvolumetheory,whichwasdevelopedbyMinkowski,Aleksandrov, Fenchel, and many others, centers around the study of geometric functionals of con- vex bodies and the differentials of these functionals.Usually, the differentials of these functionals produce new geometric measures. This theory depends heavily onanalytic tools such as the cosine transform on the unit sphere Sn−1 and Monge-Ampère type equations.
AMinkowskiproblemisacharacterizationproblemforageometricmeasuregenerated by convex bodies: It asks for necessary and sufficient conditions in order that agiven measurearisesasthemeasuregeneratedbyaconvexbody.ThesolutiontoaMinkowski problem, in general, amountsto solvinga fully nonlinear partial differential equation.
The study of Minkowskiproblems hasalonghistory andstrong influenceon both the Brunn-Minkowski theory and fully nonlinear partial differential equations. For details, see, e.g.,[45,Chapter8].
The Lp Minkowski problem for volume was originated in the 90s of last century, and it significantly generalized the classical Minkowski problem and was intensively investigated.
1.1. Lp surfacearea measuresandtheLp Minkowskiproblemforvolume
TheLpBrunn-Minkowskitheory(see,e.g.,[45,Sections9.1and9.2])isanextensionof theclassicalBrunn-Minkowskitheory,inwhichtheLp surfaceareameasureintroduced byLutwak[37] is oneofthemostfundamentalnotions.
LetKbeaconvexbodyinRn withtheorigininitsinteriorandp∈R.ItsLp surface area measureSp(K,·) isafinite Borelmeasure onSn−1,definedforBorelω⊆Sn−1 by
Sp(K, ω) =
x∈g−K1(ω)
(x·gK(x))1−pdHn−1(x),
where Hn−1 is the (n−1)-dimensional Hausdorff measure; gK : ∂K → Sn−1 is the Gauss map defined on the set ∂K of those points of ∂K that have a unique outer normal. Alternatively,Sp(K,·) canbe definedby
Sp(K, ω) =
ω
h1−pK (u)dS(K, u), (1.1)
where hK :Rn→R,hK(x)= max{x·y:y∈K}isthesupportfunctionofK;dS(K,·) is theclassicalsurfacearea measureofK.
Tracingthesource,theLpsurfaceareameasureresultedfromthedifferentialofvolume functional ofLp combinations ofconvexbodies.
In 1962, Firey [28] introduced the notionof Lp sum of convex bodies. Let K,L be convexbodieswiththeoriginintheirinteriorsand1p<∞.TheirLpsumK+pListhe compactconvexsetwithsupportfunctionhK+pL= (hpK+hpL)1/p.Fort>0,theLpscalar
multiplicationt·pKisthesett1/pK.NotethatK+1L=K+L={x+y:x∈K,y∈L} istheMinkowskisum ofK andL.
Using theLp combination,Lutwak [37] established thefollowing Lp variational for- mula
dV(K+pt·pL) dt
t=0+
= 1 p
Sn−1
hpL(u)dSp(K, u), (1.2)
where V is the n-dimensional volume, i.e., Lebesgue measure in Rn. When p = 1, it reducesto thecelebratedAleksandrov variational formula
dV(K+tL) dt
t=0+
=
Sn−1
hL(u)dS(K, u). (1.3)
Theintegralin(1.3),dividedbythefactor n,is calledthefirst mixedvolume V1(K,L) ofK andL.Thatis,
V1(K, L) = 1 n
Sn−1
hL(u)dS(K, u),
whichisageneralizationofthewell-knownvolumeformula V(K) = 1
n
Sn−1
hK(u)dS(K, u). (1.4)
The Lp Minkowski problem forvolume. Suppose μ is a finite Borel measure on Sn−1 andp∈R.Whatare thenecessary andsufficientconditions on μsuchthat μis theLp
surfaceareameasure Sp(K,·)of aconvex bodyK inRn?
TheL1 Minkowskiproblem is preciselythe classicalMinkowskiproblem.Morethan a century ago, Minkowski himself [42] solved this problem when the given measure is either discrete or has a continuous density. Aleksandrov [2,3] and Fenchel-Jessen [27]
independently solved the problem for arbitrary measures: If μ is not concentrated on anygreatsubsphereofSn−1,thenμisthesurfaceareameasureofaconvexbodyifand onlyifitscentroidisat theorigino; i.e.,
Sn−1udμ(u)=o.
RecallthattheLpMinkowskiproblemforvolumeforp<1 waspublicizedbyaseries oftalks by ErwinLutwak inthe1990’s, and appearedinprintinChou and Wang[22]
for the first time. The case p = 0 isthe so called logarithmic Minkowski problem. In [11],theauthors posedthesubspaceconcentrationconditionand completelysolvedthe even logarithmic Minkowski problem. Additional references regarding the logarithmic Minkowski problem can be found in, e.g., [7–10,20,33,46–48,50]. If 0 < p < 1, the Lp Minkowski problem is essentially solved by Chen, Li and Zhu [19]. See also [12,51],
for more details. It is worth mentioning that in the very recent work [6], the authors discussed thecase −n< p< 1 for an absolutely continuous measure and provided an almost optimalsufficientconditionforthecase0< p<1.
Since for strictly convex bodies with smooth boundaries,the density of the surface area measure withrespect to the Lebesguemeasure is justthe reciprocalof theGauss curvature of closed convex hypersurface, analytically, the classical Minkowskiproblem is equivalent to solving a Monge-Ampère equation. Establishing the regularity of the solutionisdifficultandhasledtoalongseriesofhighlyinfluentialworks,see,e.g.,Lewy [36],Nirenberg[43],Cheng-Yau[21],Pogorelov[44], Caffarelli[13,14].
By now, the Lp Minkowski problem for volume has been investigated and achieved great developments. See,e.g.,[4,11,17,18,22,32,34,37–39,41,46,50,51]. Its solutionshave beenappliedtoestablishsharpaffineisoperimetricinequalities,suchastheaffineMoser- Trudinger and theaffine Morrey-Sobolev inequalities, the affine Lp Sobolev-Zhang in- equality,etc. See,e.g.,[40,49],formoredetails.
1.2. Lp p-capacitarymeasuresandtheLp Minkowskiproblem forp-capacity
Without adoubt,theMinkowskiproblemforelectrostaticp-capacityisanextremely important variant among Minkowskiproblems. Recall thatfor 1<p< n, the electro- staticp-capacityofacompactsetK inRn isdefinedby
Cp(K) = inf
Rn
|∇u|pdx:u∈Cc∞(Rn) anduχK ,
where Cc∞(Rn) denotesthe set ofsmooth functionswith compactsupports,and χK is the characteristic function of K. The quantity C2(K) is the classical electrostatic (or Newtonian) capacityofK.
For convex bodies K and L, via the variation of capacity functional C2(K), there appearstheclassicalHadamardvariationalformula
dC2(K+tL) dt
t=0+
=
Sn−1
hL(u)dμ2(K, u) (1.5)
and itsspecialcase,thePoincarécapacityformula C2(K) = 1
n−2
Sn−1
hK(u)dμ2(K, u). (1.6)
Here,μ2(K,·) iscalled theelectrostaticcapacitary measureofK.
Inhiscelebratedarticle[35],JerisonpointedouttheresemblancebetweenthePoincaré formula(1.6) andthevolumeformula(1.4) andalsoaresemblancebetweentheirvaria- tionalformulas (1.5) and(1.3). Thus,he initiatedto study theMinkowski problemfor
electrostatic capacity: Given afinite Borelmeasure μon Sn−1,what are the necessary andsufficientconditionsonμsuchthatμistheelectrostaticcapacitarymeasureμ2(K,·) ofa convexbodyK in Rn?
Jerison[35] solved, infull generality,theMinkowskiproblem forcapacity.Heproved thenecessaryandsufficientconditionsforexistenceofasolution,whichareunexpectedly identicaltothecorrespondingconditionsintheclassicalMinkowskiproblem.Uniqueness wassettledbyCaffarelli,JerisonandLieb[16].Theregularitypartoftheproofdepends ontheideasofCaffarelli[15] fortheregularityofsolutionstotheMonge-Ampèreequa- tion.
Jerison’s workinspired muchsubsequent research onthis topic.In [24], the authors extended Jerison’s work to electrostatic p-capacity, 1 < p < n, and established the Hadamardvariationalformulaforp-capacity,
dCp(K+tL) dt
t=0+
= (p−1)
Sn−1
hL(u)dμp(K, u) (1.7)
andthereforethePoincarép-capacityformula Cp(K) = p−1
n−p
Sn−1
hK(u)dμp(K, u). (1.8)
Here,thenewmeasure μp(K,·) iscalled theelectrostaticp-capacitarymeasureofK.
Naturally,theMinkowskiproblem forp-capacitywasposed[24]:Given afiniteBorel measureμonSn−1,whatarethenecessaryandsufficientconditionsonμsuchthatμis thep-capacitarymeasureμp(K,·)of aconvexbody K inRn?
In [24], the authors proved the uniqueness of the solution when 1 < p < n, and existenceandregularitywhen 1<p<2.Veryrecently,theexistencefor2<p< nwas solvedbyM.Akman,J.Gong,J.Hineman,J.Lewis,andA. Vogel[1].
Inspiredby thedeveloped Lp Minkowski problem forvolume, D.Zou and G.Xiong [52] initiatedto researchthefollowing Lp Minkowskiproblemforp-capacitarymeasure.
Letp∈Rand1<p< n.ForaconvexbodyKinRnwiththeorigininitsinterior,its Lp p-capacitary measureμp,p(K,·) isafinite Borel measureon Sn−1,defined forBorel ω⊆Sn−1 by
μp,p(K, ω) =
ω
h1−pK (u)dμp(K, u).
Similar to the Lp surface area measure Sp(K,·), μp,p(K,·) is also resulted from the variation of p-capacity functional of Lp sumof convex bodies. Specifically, if K,L are convexbodiesinRn withtheoriginintheirinteriors,thenfor1p<∞,
dCp(K+pt·pL) dt
t=0+
= p−1 p
Sn−1
hpL(u)dμp,p(K, u).
TheLpMinkowskiproblemforp-capacity.SupposeμisafiniteBorelmeasureonSn−1, p∈R and1<p< n.What arethenecessary andsufficientconditionsonμsuchthatμ is theLp p-capacitarymeasure μp,p(K,·)of aconvexbody K inRn?
In[52], ZouandXiongcompletely solvedthecasewhenp>1 and1<p< n.Inthis article, wefocusonthecase0< p<1 and1<p<2,andprovethefollowing.
Theorem1.1. LetμbeafinitepositiveBorelmeasureonSn−1whichisnotconcentrated on any closed hemisphere, 0< p<1and1<p<2. Supposeμ isdiscreteandsatisfies μ({−u})= 0whenever μ({u})>0,foru∈Sn−1.
(1)Ifp+p=n,thenthereexistsann-dimensionalpolytopePsuchthatμp,p(P,·)=μ;
(2)If p+p=n,thenthereexistsann-dimensionalpolytopeP andaconstantλ>0 suchthatλμp,p(P,·)=μ.
This articleis organized as follows. In Section2, we collect somebasic factson the theory ofconvex bodies.InSection3, westudy anextremalproblem undertranslation transforms. After clarifying the relationship between this extremal problem and our concerned Lp Minkowskiproblemfor capacityinSection4,we presenttheproofof the main resultinSection5.
2. Preliminaries
2.1. Basicsof convexbodies
Forquickreference,wecollectsomebasicfactsonthetheoryofconvexbodies.Good referencesare thebooksbyGardner [29],Gruber[31] andSchneider[45].
Write x·y for thestandardinner product ofx,y∈Rn.Let B be the standardunit ball of Rn. Denote byKn theset of convexbodies inRn, andbyKno theset ofconvex bodieswiththeoriginointheirinteriors.
Kn is often equipped with the Hausdorff metric δH, which is defined for compact convex setsK,Lby
δH(K, L) = max{|hK(u)−hL(u)|:u∈Sn−1}.
ForcompactconvexsetsKandL,theyaresaidtobehomothetic,ifK=sL+x,for somes>0 andx∈Rn.Thereflection ofK is−K={−x:x∈K}.
Write intK and bdK for the interior and boundaryof aset K, respectively. Write relintK and relbdK for the relative interior and relative boundary of K, that is, the interior andboundaryofK relativeto itsaffinehull,respectively.
DenotebyC(Sn−1) thesetofcontinuousfunctionsdefinedonSn−1,whichisequipped with themetric induced bythe maximal norm. Write C+(Sn−1) for the set of strictly positivefunctionsinC(Sn−1).Fornonnegativef,g∈C(Sn−1) andt0,define
f+pt·g=
fp+tgp1/p
,
where, without confusion and for brevity, we omit the subscript p under the dot thereafter. If in addition f > 0 and g is nonzero, the definition holds when t >
− maxminSn−1f
Sn−1g
1/p
.
Fornonnegativef ∈C(Sn−1),define [f] =
u∈Sn−1
{x∈Rn :x·uf(u)}.
Theset iscalled theAleksandrov body (alsoknownasthe Wulffshape)associated with f. Obviously, [f] is acompact convex set containing the origin.For acompact convex setcontainingtheorigin,sayK, wehaveK= [hK].Iff ∈C+(Sn−1),then[f]∈ Kno.
TheAleksandrov convergence lemma reads: If thesequence fj
j ⊆C+(Sn−1) con- vergesuniformlyto f ∈C+(Sn−1),thenlimj→∞[fj]= [f].
ForK∈ Kn andu∈Sn−1,thesupporthyperplaneH(K,u) isdefinedby H(K, u) ={x∈Rn:x·u=h(K, u)}.
Thehalf-spaceH−(K,u) inthedirectionuisdefinedby H−(K, u) ={x∈Rn:x·uh(K, u)}. Thesupportset F(K,u) inthedirectionuisdefinedby
F(K, u) =K∩H(K, u).
Supposethe unitvectors u1,. . . ,uN are notconcentrated onany closed hemisphere ofSn−1,N n+ 1.LetP(u1,. . . ,uN) betheset withP ∈P(u1,. . . ,uN) suchthatfor fixeda1,. . . ,aN 0,
P= N k=1
{x∈Rn:x·ukak}.
Obviously,forP ∈P(u1,. . . ,uN),PhasatmostN facets,i.e.,(n−1)dimensionalfaces, and the outer normals of P are a subset of {u1,. . . ,uN}. Let PN(u1,. . . ,uN) be the subsetofP(u1,. . . ,uN) suchthatapolytopeP ∈PN(u1,. . . ,uN),ifP ∈P(u1,. . . ,uN) andP hasexactlyN facets.
2.2. Basicfacts onp-capacity
Thispartlists somenecessaryfactsonp-capacity.For moredetails,see,e.g.,[24,26, 35].
Let 1<p< n.Thep-capacityCp hasthefollowing properties.First,itisincreasing with respect to set inclusion; that is, if E ⊆ F, then Cp(E) Cp(F). Second, it is positively homogeneousofdegree (n−p),i.e., Cp(sE)=sn−pCp(E),for s>0. Third, itis rigidmotioninvariant, i.e.,Cp(gE+x)=Cp(E),forx∈Rn andg∈O(n).
Let K ∈ Kn. Thep-capacitarymeasure μp(K,·) has the following properties. First, it is positivelyhomogeneousofdegree (n−p−1),i.e., μp(sK,·)=sn−p−1μp(K,·),for s > 0. Second, it is translation invariant, i.e., μp(K+x,·) = μp(K,·), for x ∈ Rn. Third,itscentroidisattheorigin,i.e.,
Sn−1udμp(K,u)=o.Moveover,itisabsolutely continuous withrespectto thesurfaceareameasure S(K,·).
For convex bodiesKj,K ∈ Kn, j ∈N,ifKj →K ∈ Kn, thenμp(Kj,·)→μp(K,·) weakly,asj → ∞.Thisimportantfactwasprovedin[24,p.1550].
LetK ∈ Kno andf ∈C(Sn−1).There isat0>0 suchthathK+tf ∈C+(Sn−1),for
|t|< t0.TheAleksandrov body[hK+tf] is continuousint∈(−t0,t0). TheHadamard variationalformulaforp-capacity(see[24,Theorem 1.1])statesthat
dCp([hK+tf]) dt
t=0
= (p−1)
Sn−1
f(u)dμp(K, u). (2.1)
ForK,L∈ Kn,themixedp-capacityCp(K,L) (see[24,p.1549])isdefinedby
Cp(K, L) = 1 n−p
dCp(K+tL) dt
t=0+
= p−1 n−p
Sn−1
hL(u)dμp(K, u). (2.2)
When L = K, it reduces to the Poincaré p-capacity formula (1.8). From the weak convergence ofp-capacitarymeasures,itfollowsthatCp(K,L) iscontinuousin(K,L).
Thep-capacitaryBrunn-Minkowskiinequality,provedbyColesanti-Salani[25],reads:
IfK,L∈ Kn, then
Cp(K+L)n−p1 Cp(K)n−p1 +Cp(L)n−p1 , (2.3) with equality if and only ifK and L are homothetic.When p= 2,the inequality was firstestablishedbyBorell[5],andtheequalityconditionwasshownbyCaffarelli,Jerison and Lieb[16]. For more details, see, e.g., Colesanti [23] and Gardner and Hartenstine [30].
Theinequality(2.3) isequivalenttothefollowing p-capacitaryMinkowskiinequality Cp(K, L)Cp(K)n−p−1Cp(L), (2.4) with equalityifand onlyifK andL arehomothetic.See[24,p.1549] formoredetails.
2.3. Basicfacts onAleksandrov bodies
Fornonnegativef ∈C(Sn−1),define
Cp(f) =Cp([f]).
Obviously,Cp(hK)=Cp(K),foracompactsetK thatcontainstheorigin.
BytheAleksandrovconvergencelemmaandthecontinuityofCponKn,thefunctional Cp:C+(Sn−1)−→(0,∞) iscontinuous.
Let0< p<∞and1<p< n.ForK∈ Kno andnonnegativef ∈C(Sn−1),define Cp,p(K, f) = p−1
n−p
Sn−1
f(u)phK(u)1−pdμp(K, u).
Forbrevity,writeCp(K,f) forC1,p(K,f).Obviously,Cp,p(K,hK)=Cp(K).
Lemma2.1. Let0< p<∞and1<p< n.If f ∈C+(Sn−1),then Cp,p([f], f) =Cp([f]) =Cp(f).
Proof. Notethath[f]f.AbasicfactestablishedbyAleksandrovisthath[f] =f,a.e., with respect to S([f],·). That is, S([f],{h[f] < f}) = 0. Since μp([f],·) is absolutely continuous with respect to S([f],·), it follows thatμp([f],{h[f] < f})= 0.Combining thisfactandtheinequalityh[f] f, itfollowsthat
Cp,p([f], f)−Cp(f) = p−1 n−p
h[f]<f
(fp−hp[f])h1[f]−pdμp([f],·) = 0,
asdesired. 2
Lemma2.2. Let0< p<∞ andμ be afinite positive Borelmeasureon Sn−1,which is notconcentratedonanyclosedhemisphere.IfQisacompactconvexsetinRncontaining theoriginanddimQ1,then0<
Sn−1hpQdμ<∞.
Proof. That the integral is finite is clear, since μ is finite and hQ is nonnegative and bounded. To prove thepositivity of the integral, we cantake aline segment Q0 ⊆ Q, whichisoftheform Q0=l{tu0: 0t1},with0< l <∞andu0∈Sn−1.Since μis notconcentratedonanyclosedhemisphere,itimpliesthatμ({u∈Sn−1:u·u0>0})>0.
Therefore,
Sn−1
hpQdμ
Sn−1
hpQ0dμ=l
{u∈Sn−1:u·u0>0}
(u·u0)pdμ(u)>0,
asdesired. 2
Lemma2.3.LetI⊆Rbeanintervalcontaining0initsinterior,andletht(u)=h(t,u): I×Sn−1−→(0,∞)be continuous,suchthat theconvergencein
h(0, u) = lim
t→0
h(t, u)−h(0, u) t is uniformonSn−1.Then
dCp(ht) dt
t=0
= (p−1)
Sn−1
h(0, u)dμp([h0], u). (2.5)
Lemma 2.4. The convergence limi→∞Ki =K inKn isequivalent tothefollowing con- ditions takentogether:
(1) each pointin K isthelimit ofasequence xi
i withxi∈Ki fori∈N;
(2) the limit of any convergent sequence xij
j with xij ∈Kij for j ∈N belongs to K.
Fortheproofoftheabovetwolemmas,see[24,Theorem5.2] and[45,Theorem1.8.8], respectively.
3. Anextremalproblem forFp(Q,x)under translationtransforms
Supposec1,. . . ,cN ∈(0,∞) andtheunitvectorsu1,. . . ,uN arenotconcentratedon any closedhemisphere ofSn−1.Let
μ= N k=1
ckδuk(·)
be the discrete measure on Sn−1. For Q ∈ P(u1,. . . ,uN) and 0 < p < 1, define the functional Fp(Q,·):Q−→Rby
Fp(Q, x) = N k=1
ck(hQ(uk)−x·uk)p. (3.1) WeshowthereexistsauniquepointxQ∈intQsuchthatFp(Q,x) attainsthemaximum.
Lemma 3.1. Let Q ∈ P(u1,. . . ,uN) and 0 < p < 1. Then there exists a unique point xQ∈relintQsuchthat
Fp(Q, xQ) = max
x∈QFp(Q, x).
In addition,if Qisan n-dimensionalpolytope,then
N k=1
ck(hQ(uk)−xQ·uk)p−1uk=o. (3.2) Proof. First, we provethe uniqueness of the maximal point. Assume x1,x2 ∈ relintQ and
Fp(Q, x1) =Fp(Q, x2) = max
x∈QFp(Q, x).
Usingthedefinition(3.1),theJenseninequalityandtheaboveassumption,wehave Fp(Q,1
2(x1+x2)) = N k=1
ck(hQ(uk)−1
2(x1+x2)·uk)p
= N k=1
ck(1
2(hQ(uk)−x1·uk) +1
2(hQ(uk)−x2·uk))p
1
2 N k=1
ck(hQ(uk)−x1·uk)p+1 2
N k=1
ck(hQ(uk)−x2·uk)p
=1
2Fp(Q, x1) +1
2Fp(Q, x2)
= max
x∈QFp(Q, x).
SinceQisconvex, 12(x1+x2)∈Q.Sotheequalityinthethirdlineholds.Bytheequality conditionoftheJenseninequality,wehave
hQ(uk)−x1·uk =hQ(uk)−x2·uk, fork= 1, . . . , N.
Thatis,
x1·uk =x2·uk, fork= 1, . . . , N.
Since the unit vectors u1,. . . ,uN are not concentrated on any closed hemisphere, it followsthatx1=x2,whichproves theuniqueness.
Second,weprovetheexistenceofthemaximalpoint.SinceFp(Q,x) iscontinuousin x∈Q andQ iscompact, so Fp(Q,x) attains itsmaximum at apointof Q,say xQ. In thefollowing,weprovexQ ∈relintQ.
AssumexQ∈relbdQ,theboundaryofQrelativeto itsaffinehull.Fixy0∈relintQ.
Letu0= |yy0−xQ
0−xQ|. Thenforsufficiently smallδ >0,it followsthatxQ+δu0 ∈relintQ.
Next,weaimtoshowthat
Fp(Q, xQ+δu0)−Fp(Q, xQ)
= N k=1
ck(hQ(uk)−xQ·uk−δu0·uk)p− N k=1
ck(hQ(uk)−xQ·uk)p
ispositive,whichwillcontradictthemaximalityofFpatxQ.Consequently,xQ∈relintQ.
Forthis aim,wedivide{u1,. . . ,uN}into threepartsand let
U1={uk:xQ·uk=hQ(uk), uk·u0= 0, k∈ {1, . . . , N}}, U2={uk:xQ·uk=hQ(uk), uk·u0= 0, k∈ {1, . . . , N}}, U3={uk:xQ·uk< hQ(uk), k∈ {1, . . . , N}}.
WefirstclaimU1isnonempty.
AssumeU1 isempty.SincexQ∈relbdQ,thereexists aui0 ∈ {u1,. . . ,uN}suchthat xQ·ui0 =hQ(ui0).So,ui0 ∈U2,and thereforeU2 isnonempty.Inlightofthefactthat u1,. . . ,uN arenotconcentratedonanyclosedhemisphere,U3 isnonempty.
Choose apointxQ−δu0 forδ >0.Then,xQ−δu0∈/Q.
On onehand,for anyuk ∈U2, itfollowsthat(xQ−δu0)·uk =xQ·uk−δu0·uk = hQ(uk).Meanwhile,foranyuk ∈U3,sincexQ·uk < hQ(uk),forsufficientlysmallδ >0,
(xQ−δu0)·uk =xQ·uk−δu0·uk < hQ(uk).
Hence,
xQ−δu0∈
uk∈U2∪U3
x∈Rn:x·uk hQ(uk)
= N k=1
x∈Rn:x·uk hQ(uk)
=Q.
Thatis,xQ−δu0∈Q.Thisis acontradiction.Hence,U1isnonempty.
Thus,
Fp(Q, xQ+δu0)−Fp(Q, xQ)
=
uk∈U1∪U2∪U3
ck
(hQ(uk)−xQ·uk−δu0·uk)p−(hQ(uk)−xQ·uk)p
=
uk∈U1
ck(−δu0·uk)p
+
uk∈U3
ck
(hQ(uk)−xQ·uk−δu0·uk)p−(hQ(uk)−xQ·uk)p
uk∈U1
ck(−δu0·uk)p
−
uk∈U3
ck(hQ(uk)−xQ·uk−δu0·uk)p−(hQ(uk)−xQ·uk)p.
SinceU1 is nonempty,andxQ+δu0∈relintQforsufficiently smallδ >0,itfollows thatforany uk ∈U1,
−δu0·uk=hQ(uk)−(xQ+δu0)·uk >0.
So,
uk∈U1
ck(−δu0·uk)p>0. (3.3) Let
C= min
uk∈U3
(hQ(uk)−xQ·uk). Thenforanyuk∈U3,itfollowsthat
hQ(uk)−xQ·uk min
uk∈U3
(hQ(uk)−xQ·uk) =C >0.
So,forsufficientlysmallδ >0,wehave
hQ(uk)−xQ·uk−δu0·uk C 2 >0.
Bytheconcavityoftp for0< p<1,wehave
(hQ(uk)−xQ·uk−δu0·uk)p−(hQ(uk)−xQ·uk)pp C
2
p−1−δu0·uk.
So,
uk∈U3
ck(hQ(uk)−xQ·uk−δu0·uk)p−(hQ(uk)−xQ·uk)p
≤δp C
2
p−1
uk∈U3
cku0·uk.
(3.4)
Thus,accordingto(3.3) and(3.4),itfollows that Fp(Q, xQ+δu0)−Fp(Q, xQ)
uk∈U1
ck(−δu0·uk)p−δp C
2
p−1
uk∈U3
cku0·uk
=δp
uk∈U1
ck(−u0·uk)p−δ1−pp C
2
p−1
uk∈U3
cku0·uk
>0, forsufficientlysmallδ >0.SoxQ ∈relintQ.Theexistenceisproved.
Finally,weprove(3.2).SinceFp(Q,x) attainsitsmaximumattheinteriorpointxQ, we have
0 = ∂Fp(Q, x)
∂xi
x=xQ
= N k=1
ckp(hQ(uk)−xQ·uk)p−1(−uk,i),
fori= 1,. . . ,n, wherex= (x1,. . . ,xn)T anduk= (uk,1,. . . ,uk,n)T.Thatis, N
k=1
ck(hQ(uk)−xQ·uk)p−1uk=o, as desired. 2
From now on,we usexQ to denote themaximal pointof thefunctional Fp(Q,x) on Q.
Lemma 3.2.Suppose Qi ∈P(u1,. . . ,uN)andQi→Q,asi→ ∞.Then xQi →xQ and Fp(Qi, xQi)→Fp(Q, xQ), asi→ ∞. Proof. SinceQi→Q,itfollowsthatforsufficientlylargei,
xQi ∈Qi⊆Q+B.
So, xQi
i isabounded sequence.Let xQij
j beaconvergentsubsequenceof xQi
i. AssumexQij →x,butx =xQ. ByLemma2.4,itfollowsthatx ∈Q.Hence,
Fp(Q, x)< Fp(Q, xQ).
From thecontinuityofFp(Q,x) inQand x,itfollows that
j→∞lim Fp(Qij, xQij) =Fp(Q, x).
Meanwhile, byLemma2.4,for xQ ∈Q,there existsayij ∈Qij suchthatyij →xQ. Hence,
j→∞lim Fp(Qij, yij) =Fp(Q, xQ).
So,
jlim→∞Fp(Qij, xQij)< lim
j→∞Fp(Qij, yij). (3.5)
However,forany Qij,wehave
Fp(Qij, xQij)Fp(Qij, yij).
So,
jlim→∞Fp(Qij, xQij) lim
j→∞Fp(Qij, yij),
whichcontradicts(3.5).Thus,xQij →xQ,andthereforexQi →xQ.Fromthecontinuity ofFp, itfollowsthat
Fp(Qi, xQi)→Fp(Q, xQ).
Thiscompletestheproof. 2
Lemma3.3. SupposeQ∈P(u1,. . . ,uN).Then (1)Fp(Q+y,xQ+y)=Fp(Q,xQ), fory∈Rn; (2)Fp(λQ,xλQ)=λpFp(Q,xQ), forλ>0.
Proof. From thedefinition(3.1),itfollowsthat Fp(Q+y, xQ+y) = max
z∈Q+yFp(Q+y, z)
= max
z−y∈Q
N k=1
ck(hQ+y(uk)−z·uk)p
= max
z−y∈Q
N k=1
ck(hQ(uk)−(z−y)·uk)p
= max
x∈Q
N k=1
ck(hQ(uk)−x·uk)p
=Fp(Q, xQ).
Similarly,
Fp(λQ, xλQ) = max
z∈λQFp(λQ, z)
= maxz
λ∈Q
N k=1
ck(hλQ(uk)−z·uk)p
=λpmaxz
λ∈Q
N k=1
ck hQ(uk)−z λ·ukp
=λpmax
x∈Q
N k=1
ck(hQ(uk)−x·uk)p
=λpFp(Q, xQ), as desired. 2
4. Anextremalproblem relatedtotheLp capacitaryMinkowskiproblem
Suppose0< p<1,1<p<2,andμisthediscretemeasureonSn−1suchthat μ=
N k=1
ckδuk(·),
whereN n+ 1,ck >0,andu1,. . . ,uN arenotconcentratedonanyclosedhemisphere.
Inthissection, wefirststudythefollowingextremalproblem inf{max
x∈QFp(Q, x) :Q∈P(u1, . . . , uN), Cp(Q) = 1}, (4.1) and then demonstrate that its solution is precisely the solution to our concerned Lp
Minkowskiproblem forp-capacity.
Lemma 4.1. SupposeP isan n-dimensionalpolytopewith normal vectorsu1,. . . ,uN.If P isthesolutiontoproblem (4.1) andxP =o,then
λh1P−pdμp(P,·) =dμ, where λ=n−pp−1
N k=1
ckhpP(uk).
Proof. Forδ1,. . . ,δN >0 andsufficientlysmall|t|>0,let
Pt={x:x·ukhP(uk) +tδk, k= 1, . . . , N} and
α(t)Pt=Cp(Pt)−n−1pPt.
Then, Cp(α(t)Pt)= 1, α(t)Pt∈PN(u1,. . . ,uN) andα(t)Pt→P,ast→0.
Forbrevity, letx(t)=xα(t)Pt.By(3.2) ofLemma3.1,itfollowsthat N
k=1
ck(α(t)hPt(uk)−x(t)·uk)p−1uk,i= 0, fori= 1, . . . , n,
whereuk= (uk,1,. . . ,uk,n)T.Lett= 0.ThenP0=P, α(0)= 1,x(0)=oand N
k=1
ckhpP−1(uk)uk,i = 0, fori= 1, . . . , n. (4.2)
Wefirstshowx(t)
t=0 exists.Let yi(t, x1, . . . , xn) =
N k=1
ck[α(t)hPt(uk)−(x1uk,1+. . .+xnuk,n)]p−1uk,i,
fori= 1,. . . ,n.Then
∂yi
∂xj
(0,...,0)
= N k=1
(1−p)ckhpP−2(uk)uk,iuk,j. So,
∂y
∂x
(0,...,0)
n×n
= N k=1
(1−p)ckhp−2P (uk)ukuTk.
Sinceu1,. . . ,uN arenotconcentratedonanyclosedhemisphere,forx∈Rn withx=o, thereexists aui0 ∈ {u1,. . . ,uN}suchthatui0·x= 0.Thus,
xT N
k=1
(1−p)ckhpP−2(uk)ukuTk
x
= N k=1
(1−p)ckhp−2P (uk)(x·uk)2 (1−p)ci0hp−2P (ui0)(x·ui0)2>0,
whichimpliesthat
∂y
∂x
(0,...,0)
n×n
ispositivelydefinite.Bytheinversefunctiontheo- rem,itfollowsthatx(0)= (x1(0),. . . ,xn(0)) exists.
Now, we can finish the proof. Since the functional Fp attains its minimum at the polytopeP,from(2.5) and(4.2),wehave
0 =1 p
dFp(α(t)Pt, x(t)) dt
t=0
= N j=1
cjhpP−1(uj)
hP(uj)(− 1
n−p)dCp(Pt) dt
t=0
+δj−x(0)·uj
