HAL Id: hal-03000881
https://hal.archives-ouvertes.fr/hal-03000881
Submitted on 12 Nov 2020
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Regular domains and surfaces of constant Gaussian curvature in three-dimensional affine space
Xin Nie, Andrea Seppi
To cite this version:
Xin Nie, Andrea Seppi. Regular domains and surfaces of constant Gaussian curvature in three- dimensional affine space. Analysis & PDE, Mathematical Sciences Publishers, In press. �hal-03000881�
REGULAR DOMAINS AND SURFACES OF CONSTANT GAUSSIAN CURVATURE IN THREE-DIMENSIONAL AFFINE SPACE
XIN NIE AND ANDREA SEPPI
Abstract. Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In dimension three, we show that every proper regular domain is uniquely foliated by a particular kind of surfaces with constant affine Gaussian curvature. The result is based on the analysis of a Monge-Ampère equa- tion with extended-real-valued lower semicontinuous boundary condition.
1. Introduction
The results of this paper place in the context ofAffine Differential Geometry[NS94, LSZH15], and are more precisely concerned with surfaces ofconstant affine Gaussian curvature, which can be considered at the same time as a generalization ofaffine spheresand of surfaces ofconstant Gaussian curvature, and whose study has been started in [LSC97,LSZ00,WZ11].
A convex domain is said to beproperif it does not contain any entire straight line. By solving the Dirichlet problem of Monge-Ampère equation
(1.1)
(detD2w= (−w)−n−2 inΩ, w|∂Ω= 0,
on any bounded convex domainΩ⊂ Rn, Cheng and Yau [CY77] showed that in every proper convex coneC ⊂ Rn+1 there exists a unique complete hyperbolic affine sphereΣC asymptotic to the boundary∂C with affine shape operator the identity. See also [Lof10].
On the other hand, certain convex domains in the Minkowski spaceRn,1known asdomains of dependenceorregular domains[Bon05,Bar05,BBZ11] are crucial in the study of globally hyperbolic flat spacetimes [Mes07]. Such a domain is by def- inition the intersection of the futures of null hyperplanes and is determined by a lower semicontinuous functionϕ : ∂D → R∪ {+∞}, whereD ⊂ Rn is the unit ball. Bonsante, Smillie and the second author showed in [BS17,BSS19] that every 3-dimensional proper regular domain D ⊂ R2,1 contains a unique com- plete surface with constant Gaussian curvature1which generatesD. Analytically, this amounts to the unique existence of a lower semicontinuous convex function u:D→R∪ {+∞}satisfying
(1.2)
(detD2u= (1− |z|)−2 inU :=int dom(u),
u|∂D=ϕ, |∇u(x)| →+∞asx∈Utends to∂U .
2010Mathematics Subject Classification. 53A15 (primary); 35J96, 53C42 (secondary).
1
arXiv:1903.08139v2 [math.DG] 5 Apr 2019
Here “int” stands for the interior and “dom” for the subset in the domain of an extended-real-valued function where the values are real. It is also showed in [BSS19]
thatdom(u)is exactly the convex hull ofdom(ϕ)inR2.
The hyperboloid in the future light coneC0⊂R2,1provides the simplest exam- ple of both results above: It corresponds to the functionu0(z) =−(1− |z|2)12 on D⊂ R2, which satisfiesdetD2u0 = (−u0)−4 = (1− |z|)−2 withu0|∂D = 0, hence solves both (1.1) and (1.2). Geometrically, the two results can be viewed as provid- ing different ways of deformingC0, with a canonical convex surface retained in the deformed convex domain.
Statement of main results. The goal of this paper is to unify the two results via surfaces with Constant Affine Gaussian (or Gauss-Kronecker) Curvature (CAGC) k >0. The underlying analytic problem, which generalizes both (1.1) and (1.2), is (1.3)
(detD2u=ck(−wΩ)−n−2inU :=int dom(u)⊂Ω, u|∂Ω=ϕ, |∇u(x)| →+∞asx∈U tends to∂U ,
whereck >0is a constant determined bykandn, andwΩis the solution to (1.1).
The first equation in (1.3) has been called atwo-step Monge-Ampère equation[LSC97], since the Monge-Ampère equation (1.1) is involved inwΩ. Whenn= 2, we show:
Theorem A. LetΩ⊂R2be a bounded convex domain satisfying the exterior circle con- dition andϕ:∂Ω→R∪ {+∞}be a lower semicontinuous function such thatdom(ϕ) has at least three points. Then there exists a unique lower semicontinuous convex function u: Ω→R∪{+∞}which is smooth in the interior ofdom(u)and satisfies (1.3). Moreover, dom(u)coincides with the convex hull ofdom(ϕ)inR2.
Here, theexterior circle conditionmeans for everyx0∈∂Ωthere is a round disk B⊂R2containingΩsuch thatx0∈∂B. Under this condition, the right-hand side of the first equation in (1.3) goes to+∞fast enough near∂Ω, which in turn ensures the gradient blowup property ofu, namely the last condition in (1.3).
CAGC hypersurfaces and the underlying Monge-Ampère problem (1.3) were first studied by Li, Simon and Chen in [LSC97], where they proved unique solv- ability in any dimension when∂Ωandϕare bothsmooth. Thus, one of the main novelties of this paper is the consideration of boundary valueϕwith much weaker regularity assumption and possibly with infinite values, although in this situation we have to restrict to dimensionn= 2forregularityof the solutions (c.f.[BF17] and Remark8.1below).
We shall give a more precise geometric description of the CAGC surface result- ing from the functionugiven by TheoremA. It is known that a non-degenerate hypersurfaceΣ ⊂ Rn+1 has CAGC if and only if its affine normal mappingN : Σ→ Rn+1has image in an affine sphere. Givenk > 0and a proper convex cone C ⊂ Rn+1, we callΣan affine(C, k)-hypersurface if it is locally strongly convex, has CAGCkand N(Σ) lies in a scaling of the Cheng-Yau affine sphereΣC ⊂ C mentioned earlier. We deduce from TheoremA:
Theorem B. LetC ⊂R3be a proper convex cone such that the projectivized dual cone P(C∗)⊂RP∗2satisfies the exterior circle condition. LetD ⊂R3be a properC-regular domain. Then for everyk >0there exists a unique complete affine(C, k)-surfaceΣk⊂D which generatesD. Moreover,Σkis asymptotic to the boundary ofD.
Here, aC-regular domainis defined in the same way as regular domains inRn,1 mentioned earlier, except that the role of the future light coneC0⊂Rn,1is replaced byC(see Section3.1for details). The exterior circle condition onP(C∗)is equiva- lent to theinteriorcircle condition onP(C), butP(C∗)plays a more important role in our analysis because it is essentially the convex domainΩin (1.3), while the sur- faceΣkclaimed in TheoremBis given by the graph of Legendre transform of the functionufrom TheoremA. In this regard, the gradient blowup property ofucor- responds to the completeness ofΣk, whereas the last statement of TheoremBwill be proved using the last statement of TheoremA.
TheoremBand its proof imply a classification of complete affine(C, k)-surfaces:
Corollary C. Given a constantk >0and a proper convex coneC⊂R3such thatP(C∗) satisfies the exterior circle condition, there are natural one-to-one correspondences among the following three types of objects:
(a) properC-regular domainsD⊂R3, (b) complete affine(C, k)-surfacesΣ⊂R3, and
(c) lower semicontinuous functionsϕ:∂P(C∗)→R∪ {+∞}such thatdom(ϕ)has at least three points,
where the correspondence(a)↔(b)is given by TheoremB. Moreover, givenΣfrom(b), the image of the projectivized affine conormal mappingP◦N∗: Σ→P(C∗)is the interior of the convex hull ofdom(ϕ)for the correspondingϕfrom(c).
Here, the affine conormal mappingN∗ : Σ→R∗3has image in a scaling of the affine sphereΣC∗⊂C∗dual toΣC, while the projectivizationP:R∗3\ {0} →RP∗2 gives a bijection fromΣC∗toP(C∗).
Furthermore, we obtain foliations ofC-regular domains by CAGC surfaces:
Theorem D. The family of surfaces(Σk)k>0from TheoremBis a foliation ofD. Moreover, the functionK:D→Rdefined byK|Σk = logkis convex.
This foliation has been studied in the Minkowski setting in [BBZ11,BS17,BSS19], although the convexity of the “time function”Kseems to be new except whenD is the coneC itself. WhenD = C, since everyΣk is a scaling of the Cheng-Yau affine sphereΣC, the functionKis relatively easy to understand and is actually a solution to the following Monge-Ampère equation onC:
(1.4)
(detD2K=a ebK K|∂C= +∞
(a, b >0are constants, see [Sas85, Appendix A]); whereas Cheng and Yau [CY82]
proved that (1.4) has a unique solution not only for proper convex conesC⊂R3but for any proper convex domain inRd(d≥2), and the Hessian of the solution gives a complete Riemannian metric on the domain. On aC-regular domainD ⊂ R3 which is not a translation ofC, this solution does not coincide with the function Kfrom TheoremDin general, and it is worth further investigation whetherKis smooth and gives a complete Riemannian metric.
About the exterior circle condition. Let us give more discussions about the exte- rior circle condition in all the above results, which is responsible for the gradient blowup condition in (1.3) and the completeness of the surfaceΣk from Theorem Bas mentioned. In the last part of the paper, we will produce a class of examples where this condition is not satisfied and TheoremsAandBdo not hold:
Proposition E. LetΩ⊂R2be a bounded convex domain,∆⊂Ωbe an open triangle with vertices on∂Ωandϕbe the function on∂Ωvanishing at the vertices of∆withϕ= +∞
everywhere else.
(1) IfΩsatisfies the exterior circle condition at every vertex of∆(see Figure1.1 (a)), then there exists a uniqueusatisfying(1.3)as in TheoremA.
(2) If∂Ωcontains an open line segment meeting∂∆exactly at a vertex (see Figure 1.1 (b)), then there does not existusatisfying(1.3).
(a) (b)
Figure 1.1. Ωand∆under the assumptions of Parts (1) and (2) of PropositionE, respectively.
The reason for Part (2) is that anyusatisfying (1.3) can be shown to solve the first equation in (1.3) on the triangle∆with vanishing boundary value on∂∆, and then one can show that the gradient ofudoes not blowup at the vertex of∆on the line segment because the right-hand side of the equation does not go to+∞fast enough near the segment.
Geometrically, given a proper convex coneC, any circumscribed triangular cone Tas shown in Figure1.2 (c)is aC-regular domain, and we deduce from Proposition Ethat if the radial projections ofCandTonRP2look like in Figure1.2 (a), which is dual to Figure1.1 (a), thenT is foliated by affine(C, k)-surfaces as in Theorem D; whereas in the case of Figure1.2 (b), dual to Figure1.1 (b),T is not generated by any complete affine(C, k)-surface. See Corollary8.6for details.
(a) (b) (c)
Figure 1.2. The dual pictures of Figure1.1aand1.1b, and a convex cone with a circumscribed triangular cone.
Finally, we point out that if the domainDin TheoremBis preserved by an affine action of the fundamental groupπ1(S)of a closed topological surfaceS and the linear part of the action is aHitchin representationπ1(S)→SL(3,R)preserving the coneC, then the unique existence of a CAGC 1 surface preserved by the action is already proved by Labourie [Lab07, Section 8] from a different point of view. This in turn implies the unique solvability of (1.3) for the correspondingΩandϕ. In this case,Ωis aconvex divisible set[Ben04,Ben08] and does not satisfy the exterior circle condition. Nevertheless,∂Ωandϕstill have certain regularity properties, at leastϕ isR-valued and continuous (see also [Ben04,Gui05] for regularity results on∂Ω).
Finding a simple necessary condition for unique solvability of (1.3) covering this case is a problem to be further investigated.
Despite not being covered by our main results, the case of Hitchin representa- tions has particular geometric significance and is therefore one of the motivations behind our work. This is because aC-regular domain naturally arises in this case asdomain of discontinuity. In fact, in the Minkowski setting, Mess [Mes07] showed that any isometric action ofπ1(S)onR2,1with linear part aFuchsian representation π1(S) → SO(2,1)is properly discontinuous on some regular domainD and the quotientD/π1(S) ∼= S×Ris a prototype ofmaximal globally hyperbolic flat space- times. We anticipate a similar result for affine actions with Hitchin linear parts.
However, we will not pursue surface group actions further in this paper.
Organization and methods of the paper. After reviewing backgrounds on affine differential geometry in Section2, we introduce in Section3 the main objects of this paper: We first defineC-regular domains andC-convex hypersurfaces, a natu- ral class of convex hypersurface generalizing the so-calledfuture-convexspacelike hypersurfaces in the Minkowski space, then we discuss hypersurfaces with CAGC and define affine(C, k)-hypersurfaces.
Section4introduces some tools from convex analysis, such as convex envelopes, subgradient and Legendre transformation. Then in Section5we relateC-regular domains andC-convex hypersurfaces to the setting of convex analysis, by estab- lishing some fundamental correspondences that enable to translate our geometric problems into an analytic framework.
Moving forward to the analytic setup, Section6briefly reviews some ingredients in the theory of Monge-Ampère equations, then the last two sections provide the proofs of our main results. Section7first provides the concrete formulation of the problem as in (1.3), and then obtains some fundamental local estimates, close to the boundary, on the Cheng-Yau solutions of (1.1).
We finally solve the Monge-Ampère problem (1.3) in dimension2in Section8.
While the existence and uniqueness of the Monge-Ampère equation with Dirichlet boundary condition follows from more or less standard arguments, proving the gradient blowup involved more subtle estimates, which rely on the estimates on the Cheng-Yau solutions given in Section7. Applying techniques of a similar type, we also produce the counterexamples of PropositionE.
Acknowledgments. We are grateful to Francesco Bonsante for helpful discussions and to Connor Mooney for helps with the proofs in Section8.2. The work leading to this paper was done during the first author’s visit to University of Pavia and the second author’s visit to KIAS. We would like to thank the respective institutes for their warm hospitality. The second author is member of the Italian national research group GNSAGA.
Contents
1. Introduction 1
2. Preliminaries on affine differential geometry 6
3. C-regular domains and hypersurfaces with CAGC 11
4. Preliminaries on convex analysis 15
5. Legendre transforms ofC-regular domains andC-convex hypersurfaces 23
6. Preliminaries on Monge-Ampère equations 33
7. Monge-Ampère equation for affine(C, k)-hypersurfaces 36
8. Analysis of the Monge-Ampère equation 41
References 51
2. Preliminaries on affine differential geometry
The section gives a concise review of background materials on affine differential geometry used later on.
2.1. Intrinsic data of affine immersions relative to a transversal vector field. Fix d= n+ 1≥3. For conceptual clearness, we use different notations forRdwhen different structures are under consideration:
• LetVdenote thed-dimensional real vector space endowed with a volume form. We consider the volume form as the determinantdet :Vd
V→R.
• LetV∗denote the vector space dual toV, consisting of linear forms onV.
• LetAdenote thed-dimensional affine space modeled onV, endowed with the volume form given by that onV. In other words,Ais obtained fromV by “forgetting” the origin.
In this section, by ahypersurfaceinA, we always mean a smooth embedded one, which has dimension n. Given a hypersurfacesΣ ⊂ A, Affine Differential Ge- ometry studies properties ofΣinvariant under volume-preserving affine transfor- mations ofA. To achieve this, one considers the intrinsic invariants ofΣdefined relative to a transversal vector fieldN : Σ→V. These invariants include a volume formν, a torsion free affine connection∇, a symmetric2-tensorh, a(1,1)-tensorS and a1-formτonΣ, determined as follows:
• νis theinduced volume form, defined by
ν(X1,· · · , Xn) = det(X1,· · ·, Xn, N),
Here and below,X,Y andX1,· · · , Xnare any tangent vector fields onΣ.
• ∇andhare theinduced affine connectionandaffine metric, respectively, de- termined by
DXY =∇XY +h(X, Y)N.
• Sandτare theshape operatorandtransversal connection form, determined by DXN =S(X) +τ(X)N.
Remark2.1. Following the original convention of Blaschke, in the literature a minus sign is often added to the definition ofS, which is inconvenient for our purpose.
Given an open setU ⊂ Rn, an immersionf : U → Aand a mapN : U → V transversal tof, one can define the above invariants onU by pullback. Thefunda- mental theoremof affine differential geometry, similar to the one in classical surface theory, states that these invariants satisfy certain structural equations, and con- versely whenUis simply connected, any prescribed invariants satisfying the equa- tions are realized by somefandN, unique up to volume-preserving affine trans- formations. Therefore, we call the quintuple of invariants(ν,∇, h, S, τ)theintrinsic dataof the pair(Σ, N)or(f, N).
We give below some basic facts about these invariants to keep in mind. Here, a smooth function is said to belocally strongly convexif its hessian is positive definite everywhere, while a hypersurface is said to be locally strongly convex if it can be presented locally as graphs of such functions.
- The rank and signature of the affine metrichonly depends onΣ, not on N, andΣis said to benon-degenerateifhis. Thus,his a genuine pseudo- Riemannian metric in this case.
- A locally convex hypersurface is non-degenerate if and only if it is locally strongly convex. Moreover, his a Riemannian metric if and only ifΣis locally strongly convex withNpointing towards the convex side.
- We havedτ= 0if and only ifh(X, S(Y)) =h(S(X), Y). As a consequence, ifΣis non-degenerate anddτ= 0thenShasnreal eigenvalues everywhere.
A transversal vector fieldN is said to beequi-affineif the transversal connection formτvanishes. We thus omit theτcomponent when talking about intrinsic data in this case. From the definitions ofτandS, we deduce the following basic fact:
Lemma 2.2. Givenp∈Σ, we haveτ = 0atpif and only if the image of the tangent map dNp : TpΣ → Vis contained in the tangent spaceTpΣ ⊂ TpA ∼= V. In this case, the image ofdNpequalsTpΣif and only if the shape operatorSis non-degenerate atp. 2.2. Affine normals and affine spheres. If Σis non-degenerate, there exists an equi-affine vector fieldNsuch that the induced volume formν coincides with the volume formdvolhof the affine metrich(note thatdvolhis defined using the orien- tation induced byν). The vector fieldNis unique up to sign and is called anaffine normal fieldofΣ, or anaffine normal mappingwhenN is viewed as a map fromΣto V. As before, we extend the definition to define affine normal mappingN :U →V of an immersionf :U →A.
Given an equi-affine transversal vector fieldN, ifν is a constant multiple of dvolh, one can scaleN to get an affine normal field:
Lemma 2.3. LetΣ ⊂ Abe a hypersurface with equi-affine transversal vector fieldN : Σ→Vand intrinsic data(ν,∇, h, S). Supposedvolh =aν for a constanta6= 0. Then
±|a|n+22 Nare the affine normal fields ofΣ(recall thatdim(A) =n+ 1).
Proof. This follows from the definition and the fact that if we scaleNby a constant λ6= 0, thenνandhget scaled byλand λ1, respectively.
While in general there is no privileged choice between the two affine normal fields opposite to each other, we do have one whenΣis locally strongly convex:
the one pointing towards the convex side ofΣ. In this case, we call the shape op- eratorSofΣwith respect to this choice ofN theaffine shape operatorand call the determinantdet(S) : Σ → Rtheaffine Gauss-Kronecker curvature, or simplyaffine Gaussian curvature.
Definition 2.4(Proper and hyperbolic/elliptic affine spheres). A non-degenerate hypersurfaceΣ⊂Ais called aproper affine sphereif its shape operator with respect to an affine normal field isS = λidfor a constantλ 6= 0(whereid denotes the identity endomorphism ofTΣ). This condition is equivalent to the existence of a pointo∈A, called thecenterofΣ, such thatN(p) =λ−→op(for allp∈Σ) is an affine normal field. Furthermore,Σis called ahyperbolic(resp. elliptic) affine sphere if Σis locally convex and the condition is satisfied withλ >0(resp. λ <0) for the affine normal field pointing towards the convex side ofΣ.
Remark2.5. The terminology “proper” is used here as opposed to the caseS = 0, in whichΣis known as animproperaffine sphere.
Thus, the center of a hyperbolic (resp. elliptic) affine sphereΣlies on the concave (resp. convex) side ofΣ. In the sequel, when talking about proper affine spheres in the vector spaceV, we always mean those centered at the origin0∈V.
2.3. Intrinsic data of equi-affine vector field as centro-affine immersion. A hy- persurfaceΣ⊂Vis said to becentro-affineif the position vector−→
0pof every point p∈Σis transversal toΣ. Thus, proper affine spheres inVare centro-affine.
Given a hypersurfaceΣ⊂Awith equi-affine transversal vector fieldN : Σ→V, if the shape operator of(Σ, N)is non-degenerate, then Lemma2.2implies thatNis a centro-affine immersion ofNintoV. Its intrinsic data with respect to the position vector field are given by the intrinsic data of(Σ, N)as follows:
Lemma 2.6. LetΣ ⊂ Abe a hypersurface,N : Σ → Vbe an equi-affine transversal vector field and(ν,∇, h, S)be the intrinsic data of(Σ, N). Supposedet(S)6= 0onΣand considerN as a centro-affine hypersurface immersion. Then the intrinsic data ofN with respect to its position vector field (given byNitself) is
(det(S)ν, S−1∇S, h(·, S(·)), id).
Here, the affine connection∇0 = S−1∇S is the gauge transform of∇byS−1, defined by∇0XY :=S−1∇X(S(Y))for any tangent vector fieldsX andY onΣ.
To prove Lemma2.6, we use the following framework to compute intrinsic data in coordinates, which is also needed in Section7.1below. LetU ⊂Rnbe an open set,f :U →Abe an immersion andN :U →Vbe a transversal vector field tof. Then the induced volume form of(f, N)is
(2.1) ν= det(∂1f,· · · , ∂nf, N)dx1∧ · · · ∧dxn,
where x = (x1,· · ·, xn)is the coordinate on U and ∂if(x), N(x) ∈ V ∼= Rn+1 are written in coordinates as column vectors. The other intrinsic invariants are determined by the following equality of(n+ 1)×(n+ 1)matrices of1-forms:
(2.2) d(∂1f,· · · , ∂nf, N) = (∂1f,· · ·, ∂nf, N)
A Sdx
tdx h τ
.
Here,dxrepresents the column vector of1-formst(dx1,· · ·dxn). The shape opera- torS= (Sij)and affine metrich= (hij)are written asn×nmatrices of functions on U, andAis the matrix of1-forms such that the induced affine connection∇is ex- pressed under the frame(∂1,· · · , ∂n)as∇=d+A. Note that the gauge transform of∇byS−1isS−1∇S=d+S−1AS+S−1dS.
Proof of Lemma2.6. LetU ⊂ Rn be an open set. We can replaceΣandN in the statement of the lemma by an immersionf : U → A and a mapN : U → V transversal tof, respectively.
The intrinsic data(ν,∇, h, S)of(f, N)are determined by the equalities (2.1) and (2.2) withτ= 0. The last column of (2.2) givesdN = (∂1f,· · ·, ∂nf)Sdx, hence (2.3) (∂1N,· · ·, ∂nN, N) = (∂1f,· · ·, ∂nf, N)
S 1
.
Therefore, the induced volume form ofN is
ν0= det(∂1N,· · · , ∂nN, N)dx1∧ · · · ∧dxn= det(S)ν,
as required, while the induced affine connection∇0 =d+A0 and affine metrich0 are characterized by
d(∂1N,· · ·, ∂nN, N) = (∂1N,· · ·, ∂nN, N) A0 dx
tdx h0 0
! .
By (2.3) and (2.2), the left-hand side is d(∂1N,· · ·, ∂nN, N) =d(∂1f,· · · , ∂nf, N)
S 1
+ (∂1f,· · · , ∂nf, N) dS
0
= (∂1f,· · ·, ∂nf, N)
A Sdx
tdx h 0
S 1
+
dS 0
= (∂1N,· · · , ∂nN, N) S−1
1
AS+dS Sdx
tdx hS 0
.
Comparing with the right-hand side, we get the required expressions A0=S−1AS+S−1dS, h0=hS.
2.4. Complete hyperbolic affine spheres. The following theorem classifies hyper- bolic affine spheres that arecompletein the sense that the Riemannian metric on it induced by an ambient Euclidean metric is complete (see also Remark3.4below):
Theorem 2.7 (Cheng-Yau [CY77]). For any proper convex cone C ⊂ V, there exits a unique complete hyperbolic affine sphereΣC ⊂ Casymptotic to∂C with affine shape operator the identity.
Here, ΣC beingasymptoticto ∂C means the distance from x ∈ Σto ∂C with respect to an ambient Euclidean metric tends to0asxgoes to infinity inΣ.
The theorem essentially gives all complete hyperbolic affine spheres because for any such affine sphereΣ ⊂ V, sinceΣis centro-affine, the projectivization map P:V\ {0} →RPnis a diffeomorphism fromΣto a convex domain inRPnand if we letCbe the component of the pre-imageP−1(P(Σ))containingΣ, thenCis a proper convex cone andΣis a scaling of the affine sphereΣC from the theorem.
For later use, we determine the precise relation between the scale factor and the affine shape operator ofΣas follows:
Proposition 2.8. Givenλ >0and a proper convex coneC⊂V, the scalingλΣCof the affine sphereΣCfrom Theorem2.7is the unique complete hyperbolic affine sphere asymp- totic to∂Cwith affine shape operatorλ−2(n+1)n+2 id(wheredim(V) =n+ 1).
Proof. Lethandν = dvolhbe the affine metric and induced volume form ofΣC, which are defined with respect to the affine normal field ofΣC, namely the position vector field. Lettingf : ΣC →λΣCdenote the scaling map, one can check from the definitions that the induced volume formν˜and affine metric˜hofλΣCwith respect to its own position vector field are related to the push-forwards ofνandhby
˜
ν=λn+1f∗ν, ˜h=f∗h.
So we havedvol˜h=f∗dvolh=f∗ν=λ−n−1ν˜, and Lemma2.3implies that the affine normals ofλΣCareλ−2(n+1)n+2 times its position vectors. The required expression of
affine shape operator follows.
The following fundamental examples are among the rare cases whereΣCadmits an explicit expression. Seee.g.[Lof10] for details.
Example 2.9(Hyperboloid).TheMinkowski spaceRn,1isVendowed with a bilinear form of signature(n,1)whose underlying volume form is the prescribed one. By convention, we pick a componentC0of the quadratic cone{v∈V|(v, v)<0}and callC0thefuture light coneinRn,1. The affine sphereΣC0 claimed by Theorem2.7 turns out to be the component of the two-sheeted hyperboloid{(v, v) =−1}inC0, namelyΣC0 =H:={v∈C0|(v, v) =−1}.
Example 2.10(Ţiţeica affine sphere). Let(v0,· · ·, vn)be a unimodular basis ofV.
For the simplicial coneC1:={t0v0+· · ·+tnvn|ti>0}, there is a constantΛn >0 only depending onnsuch that the affine sphere claimed by Theorem2.7is
ΣC1 ={t0v0+· · ·+tnvn|t1,· · · , tn>0, t0· · ·tn= Λn}.
2.5. Affine conormals and dual affine sphere. Given a hypersurfaceΣ⊂Awith affine normal fieldN : Σ→ V, theaffine conormalofΣatp∈ Σdual toN is the linear formN∗(p)∈V∗defined by
hN∗(p), N(p)i= 1, hN∗(p),TpΣi= 0,
where “h·,·i” is the pairing betweenVandV∗and the last equality meanshN∗(p), vi= 0for everyv∈TpΣ⊂TpV∼=V. We callN∗: Σ→V∗the affine conormal mapping dual toN.
IfΣhas non-degenerate shape operator with respect toN, so thatN is an im- mersion ofΣintoVas a centro-affine hypersurface (see Lemma2.2and Section 2.3), thenN∗: Σ→V∗is a centro-affine immersion as well and its imageN∗(Σ)is the centro-affine hypersurfaceM∗dualtoM =N(Σ), defined by
M∗={α∈V∗| there isv∈M such thathα, vi= 1,hα,TvMi= 0}.
WhenMis a proper affine sphere with affine shape operatorλid, it can be shown that the dual centro-affine hypersurfaceM∗defined above is a proper affine sphere inV∗with affine shape operatorλ−1id, soM∗is called thedual affine sphereofM. It can also be shown that for the hyperbolic affine sphereΣCfrom Theorem2.7, the dual affine sphere is exactlyΣC∗, whereC∗⊂V∗is the dual cone ofC, consisting of linear forms onVwhich take positive values onC\ {0}. It follows that the dual affine sphere ofλΣCisλ−1ΣC∗.
3. C-regular domains and hypersurfaces with CAGC
In this section, we define the main objects of study of this paper:C-regular do- mains,C-convex hypersurfaces and affine(C, k)-hypersurfaces. WhenCis the fu- ture light cone C0 in the Minkowski spaceRn,1, the first two objects are known in the literature as regular domains and future-convex spacelike hypersurfaces, re- spectively, while we show in Section3.3that affine(C0, k)-hypersurfaces are exactly C0-convex hypersurface with constant Gaussian curvature in the classical sense.
3.1. C-regular domains andC-convex hypersurfaces. As in Section2, we fixn≥ 2, letVdenote an(n+ 1)-dimensional vector space equipped with a volume form andAdenote the affine space modeled onV. By aconvex domain, we mean a convex open set, while aconvex coneinVis a convex domain invariant under positive scal- ings. A convex cone/domain isproperif it is nonempty and does not contain any entire straight line. We further introduce the following definitions and notations:
• LetHAdenote the space of allopen half-spacesofA,i.e. open subsets whose boundaries are affine hyperplanes.
• HVdenote the space of all open half-spaces ofVwith boundaries passing through the origin0∈V. Thus, there is a natural projectionHA→ HVsuch that the pre-image ofH∈ HVconsists of the translations ofH.
• Asupporting half-spaceof a convex domainDat a boundary pointp∈∂D is an open half-spaceH such thatD⊂Handp∈∂H.
• Given a convex coneC ⊂ V, we letHV(C) ⊂ HVdenote the space of all supporting half-spaces of C and letH0
V(C) ⊂ HV(C) denote the set of supporting half-spaces at boundary points other than the origin. Also put H1
V(C) :=HV(C)\ H0
V(C). See Figure3.1.
H1 H0
C
Figure 3.1. The coneC and the boundary hyperplanes of some H0 ∈ H0
VandH1 ∈ H1
V. The half-spacesH0andH1are the parts ofVabove the respective hyperplanes.
• LetH0A(C)andH1A(C)denote the pre-images ofH0V(C)andH1V(C)inHA, respectively.
Adapting terminologies from Minkowski geometry, we call elements ofH0A(C)and H1A(C) C-null and C-spacelikehalf-spaces, respectively, and call an affine hyper- planeC-null/C-spacelike if it is the boundary of aC-null/C-spacelike half-space.
WhenCis the future light coneC0in the Minkowski spaceRn,1(see Example2.9), these are just null/spacelike hyperplanes in the classical sense.
The following notion also arises from the Minkowski setting:
Definition 3.1(C-regular domains). Given a proper convex cone C ⊂ V, aC- regular domainis by definition a subsetD⊂Aof the form
D=int \
H∈H
H
!
, H ⊂ H0A(C).
Namely,Dis the interior of the intersection of a collectionHofclosedC-null half- spaces. WhenHis the set of allC-null half-spaces containing some subsetSofA, Dis called theC-regular domaingenerated byS.
Remark3.2. For the future light coneC0 ⊂ Rn,1, such domains first appeared in mathematical relativity asdomains of dependence, then the name “regular domain”
is introduced in [Bon05]. Our definition is slightly wider even forC=C0, in that regular domains in [Bon05] correspond toproperC0-regular domains under our definition. We refer to the papers cited in the introduction for the role of such domains in the study of globally hyperbolic flat spacetimes.
Note thatC-regular domains are convex domains, and the simplest examples include the empty set∅, the wholeA(corresponding toH=∅), singleC-null half- spaces, and translations of the coneCitself.
A convex hypersurfaceΣ ⊂ Ais by definition a nonempty open subset of the boundary of some convex domain, and a supporting half-space of the domain at a pointp ∈Σis referred to as a supporting half-space ofΣatp. We will study the following particular type of convex hypersurfaces, known asfuture-convex spacelike hypersurfaces whenC=C0⊂Rn,1:
Definition 3.3(C-convex hypersurfaces). Given a proper convex coneC⊂V, aC- convexhypersurface is a convex hypersurfaceΣ⊂Awhose supporting half-spaces are allC-spacelike. Σis said to be completeif it is the entire boundary of some convex domain.
Remark3.4. For alocallyconvex immersed hypersurface, there is a nontrivial rela- tion between the notion of completeness defined above and the completeness of the geodesic metric on the hypersurface induced by an ambient Euclidean metric, see [VH52]. However, since we only consider embedded globally convex hypersur- faces, these notions are the same.
As an example, for the affine sphereΣC ⊂C given by Theorem2.7, using the fact thatΣCis equi-affine and asymptotic to∂C, it can be shown that any scaling λΣCis a completeC-convex hypersurface generating the convex coneC.
In Section5, we will identify all completeC-convex hypersurfaces inAas the entire graphs of a specific class of convex functions onRn, and identifyC-regular domains as strict epigraphs of specific convex functions as well. We will also see that if a completeC-convex hypersurfaceΣis asymptotic to the boundary of aC- regular domainD(in the sense defined in Section2.4), thenΣgeneratesD, whereas the converse is not true.
3.2. Convex hypersurfaces with Constant Affine Gaussian Curvature. With the definitions from Section2in mind, by a hypersurface inAwithconstant affine Gauss- ian curvature(CAGC), we mean a non-degenerate smooth embedded hypersurface Σ ⊂ Asuch that the shape operator ofΣwith respect to an affine normal field has constant determinant. Since there are two affine normal fields opposite to each
other, the precise value of affine Gaussian curvature (i.e.the determinant) has sign ambiguity whennis odd. WhenΣis locally convex, the ambiguity is eliminated by picking the affine normals pointing towards the convex side as mentioned in Section2.2.
The following result characterizes CAGC hypersurfaces by affine normal map- pings and singles out a subclass of these hypersurfaces which we study later on:
Proposition 3.5. LetΣ⊂Abe a non-degenerate smooth hypersurface with affine normal mappingN : Σ → V. LetS be the shape operator ofΣwith respect toN and suppose det(S)6= 0onΣ, so thatNis an immersion (see Lemma2.2). Then
(1) det(S)is constant if and only ifN(Σ)is a proper affine sphere inV. In this case, an affine normal field ofN(Σ)is given by|det(S)|−n+21 times its position vectors.
(2) Further assume thatΣis locally strongly convex andNpoints towards the convex side ofΣ. ThenN(Σ)is a hyperbolic affine sphere if and only ifdet(S)is a constant and the eigenvalues ofSare all positive at every point ofΣ.
Proof. (1) Let(ν,∇, h, S)be the intrinsic data of(Σ, N), whereν = dvolh is the volume form ofhsinceN is an affine normal field. By Lemma2.6, the induced volume formν0and affine metrich0of the centro-affine immersionNwith respect to its position vector field are given byν0 = det(S)ν andh0(·,·) = h(·, S(·)). The volume form ofh0is
dvolh0 =|det(S)|12dvolh=|det(S)|12ν =±|det(S)|−12ν0.
In particular,det(S)is a constant if and only ifdvolh0 is a constant timesν0. But Lemma2.3implies thatdvolh0 is a constanta6= 0timesν0if and only|a|n+22 times the position vector field ofN(Σ)is an affine normal field, hence the required state- ments follows.
(2) The additional assumption is equivalent to the condition thathis positive definite (see Section2.1). In this case, the eigenvalues ofS are all positive if and only ifh0is positive definite as well. But the positive definiteness ofh0is equivalent to the condition thatN(Σ)is a locally convex centro-affine hypersurface with the position vector of each point pointing towards the convex side (or equivalently, 0∈Vlies on the concave side). WhenN(Σ)is a proper affine sphere, this condition means exactly thatN(Σ)is actually a hyperbolic affine sphere. The first statement
then follows from Part (1).
With the same proof, one can show a similar statement as Part (2) with “hyper- bolic” and “positive” replaced by “elliptic” and “negative”, respectively. But we are mainly interested in the situation whereN(Σ)is not merely a hyperbolic affine sphere, but also part of acompleteone discussed in Section2.4:
Definition 3.6(Affine(C, k)-hypersurfaces). LetC ⊂Vbe a proper convex cone and letk >0. Anaffine(C, k)-hypersurfaceinAis a locally strongly convex smooth hypersurfaceΣwith CAGCksuch that the affine normal mappingN : Σ→Vhas image in a complete hyperbolic affine sphere generatingC.
Given an affine(C, k)-hypersurfaceΣ⊂A, by Propositions2.8and3.5, the affine normal mappingN : Σ → Vand conormal mappingN∗ : Σ→ V∗have images in the following specific scaling of the Cheng-Yau affine spheresΣCandΣC∗(see Sections2.4and2.5), respectively:
N(Σ)⊂k2(n+1)1 ΣC, N∗(Σ)⊂k−2(n+1)1 ΣC∗.
The former is the complete hyperbolic affine sphere generatingCwith affine shape operatork−n+21 idby Proposition2.8, and the latter is dual to the former. Also note thatΣisC-convex because by Lemma2.2and the fact thatN points towards the convex side ofΣ, the supporting half-space ofΣat a pointpcoincides with that of N(Σ)atN(p)up to translation, whilek2(n+1)1 ΣCand henceN(Σ)areC-convex.
3.3. The Minkowski case. We view the Minkowski spaceRn,1 (see Example2.9 for the basic definitions) both as a vector space endowed with a bilinear form(·,·) of signature(n,1)(hence also endowed with the resulting volume form) and as the affine space modeled on this vector space, which is a flat Lorentzian manifold.
A hypersurfaceΣ ⊂ Rn,1 is said to bespacelikeif its tangent hyperplanes are spacelike (c.f. Section3.1). WhenΣisC1, it is the case if and only if the ambient Lorentzian metric restricts to a Riemannian metricgonΣ, which is called thefirst fundamental formofΣ. Note thatC0-convex hypersurfaces (see Definition3.3) are in particular spacelike.
From an affine-differential-geometric point of view, there is a natural equi-affine transversal vector field to be considered for a spacelike hypersurfacesΣ: the unit normal field, taking values in the hyperboloidH(see Example2.9). The resulting intrinsic data(ν,∇, h, S)have the following relations with the first fundamental formg:
• The induced volume formν and induced affine connection∇are the vol- ume form and Levi-Civita connection ofg, respectively.
• The affine metrich, known as thesecond fundamental formofΣ, is related to the shape operatorSthrough the relationh(·,·) =g(·, S(·)).
We call the determinantdet(S) : Σ → RtheclassicalGaussian curvature ofΣin order to distinguish with the affine Gaussian curvature. Note that whenn= 2, the Gauss equationforΣ, similar to the one for surfaces in the Euclidean3-space, states that the classical Gaussian curvature is opposite to the intrinsic curvature of the first fundamental form.
As the main result of this section, we characterize affine(C0, k)-hypersurface by classical Gaussian curvature:
Proposition 3.7. Givenk >0, affine(C0, k)-hypersurfaces in the Minkowski spaceRn,1 are exactlyC0-convex hypersurfaces with constant classical Gaussian curvaturek2n+2n+2. Proof. LetΣ⊂Rn,1be an affine(C0, k)-hypersurface andN : Σ→Rn,1be its affine normal mapping. ThenΣisC0-convex andNhas image in thek2(n+1)1 -scaling of ΣC0 =H(see Section3.2). Thus,N0:=k−2(n+1)1 Nhas images inH. By Lemma2.2, N0is a local diffeomorphism fromΣtoHand for everyp∈ Σ, the tangent space TpΣcoincides with dNp0(TpΣ) = TN0(p)Hafter translation. But the orthogonal complement of the vectorN0(p)is exactly the subspaceTN0(p)H⊂TpRn,1∼=Rn,1. Therefore,N0is the unit orthogonal normal field. The shape operatorS0ofΣwith respect toN0isS0 =k−2(n+1)1 S, hence the classical Gaussian curvature isdet(S0) = k−2(n+1)n det(S) =k2n+2n+2, as required.
Conversely, letΣ ⊂ Rn,1 be aC0-convex hypersurface with first fundamental formg and unit orthogonal normal fieldN0 : Σ → H, and let (ν0,∇0, h0, S0)be the intrinsic data of(Σ, N0). Assume the classical Gaussian curvaturedet(S0)is a
constantk0>0. Sinceν0=dvolgandh0(·,·) =g(·, S0(·)), the volume form ofh0is dvolh0 =k012dvolg=k012ν0.
By Lemma2.3,N =k0n+21 N0is the affine normal field ofΣ, hence the affine Gauss- ian curvature isdet(S) = det(k0n+21 S0) =k02n+2n+2 =:k. Thus,Σis an affine(C0, k)-
hypersurface, as required.
Remark 3.8. One can consider affine-differential-geometric properties of general (not necessarily locally convex) spacelike hypersurfacesΣ⊂Rn,1, or hypersurfaces in the Euclidean spaceEn+1. In particular, refining the argument in the second part of the above proof, it can be shown that for such aΣinRn,1orEn+1, a unit normal vector fieldN (with values inHand the unit sphere Sin the Minkowski and Euclidean cases, respectively) scaled by some functiona: Σ → R+gives an affine normal field if and only if the classical Gaussian curvature ofΣis a nonzero constant, and in this caseamust be a constant. Therefore, ifΣhas constant classical Gaussian curvature then it also has CAGC.
4. Preliminaries on convex analysis
In this section, we introduce definitions, notations and results from convex anal- ysis that we will use in the following sections.
4.1. Convex functions. We consider lower semicontinuous convex functions on Rnwith values inR∪ {+∞}and denote the space of them, excluding the constant function+∞, by
LC(Rn) :={u:Rn→R∪{+∞} |uis lower semicontinuous and convex,u6≡+∞}.
Also, it is sometimes convenient to consider the space LC(f Rn) :=LC(Rn)∪ {±∞},
where+∞and−∞are understood as constant functions. It has the property that the pointwise supremum of any family of functionsF ⊂LC(f Rn)is still inLC(f Rn) (the supremum of the empty set is−∞by convention). Members inLC(f Rn)and LC(Rn)are called “closed convex functions” and “closed proper convex functions”, respectively, in literatures on convex analysis such as [Roc70]. Here “closed” refers to the closedness of the epigraph
epi(u) :={(x, ξ)∈Rn×R|ξ≥u(x)}.
In fact,uis convex/lower semicontinuous if and only ifepi(u)is convex/closed.
Givenu∈LC(Rn), theeffective domain
dom(u) :={x∈Rn|u(x)<+∞}
is a nonempty convex subset ofRn. It is a basic fact that any R-valued convex function on an open subset of Rn is continuous (actually Lipschitz, see [Gut01, Lemma 1.1.6]), souis continuous in the interiorU := int dom(u)of its effective domain. At a boundary pointx0∈∂U,uis continuous at least along line segments inUin the sense that
t→0lim+u((1−t)x0+tx1) =u(x0) for anyx0∈∂U, x1∈U.
Indeed, by lower semicontinuity we havelim inft→0+u((1−t)x0+tx1) ≥ u(x0), while by convexity we have u((1−t)x0 +tx1) ≤ (1−t)u(x0) +tu(x1), hence
lim supt→0+u((1−t)x0+tx1) ≤ u(x0). Note thatu(x0) = +∞is allowed here.
More generally, a similar argument shows that the restriction ofuto every simplex indom(u)is continuous (see [Roc70, Theorem 10.2]).
By virtue of these continuity properties, ifUis nonempty, thenuis determined by its restriction toU:
Proposition 4.1. Given a convex domainU ⊂Rnand a convex functionu:U →R, the functionu:Rn →R∪ {+∞}defined by
u(x) :=
u(x) ifx∈U
lim infy→xu(y) ifx∈∂U
+∞ ifx∈Rn\U
is the unique element inLC(Rn)extendinguwith effective domain contained inU. The extensionuis a particular instance ofconvex envelopes, introduced in the next section for more generalu.
Proof. It is elementary to check that the epigraphepi(u)is exactly the closure of epi(u)⊂U×RinRn×R. Sinceepi(u)is convex,epi(u)is a closed convex set, hence u∈LC(Rn). To prove the uniqueness, it is sufficient to show that givenu∈LC(Rn) withU :=int dom(u)nonempty, the value ofuat anyx0 ∈∂U coincides with the liminf ofu(x)asx∈U tends tox0. This is given by
lim inf
x∈U, x→x0
u(x)≥u(x0) = lim
t→0+u((1−t)x0+tx1)≥ lim inf
x∈U, x→x0
u(x),
where the first inequality is becauseuis lower semicontinuous.
For generalu∈LC(Rn)withint dom(u)possibly empty, there is a unique affine subspaceA⊂Rn containingdom(u)such thatdom(u)has nonempty interior as a subset ofA, and one can study the restrictionu|A∈LC(A)instead. The interior of dom(u)as a subset ofAis called therelative interiorand denoted byri dom(u)(see [Roc70, Section 6]).
Proposition4.1assigns to every convex functionu:U →Ritsboundary values, namely, the restrictionu|∂U. This is fundamental in the study of Monge-Ampère equations onUbecause the Dirichlet problem of such equations asks for a convex function with prescribed boundary values. A basic fact is thatuextends continu- ously toU if and only if its boundary values are continuous. This follows from:
Proposition 4.2. Letu ∈ LC(Rn)withU := int dom(u)nonempty and letx0 ∈ ∂U be such thatu(x0) <+∞and the restriction ofuto∂U is continuous atx0. Then the restriction ofutoU is continuous atx0as well.
Proof. Suppose u|U is not continuous atx0 and pick a pointy0 ∈ U. Adding an affine functionRn → R toudoes not affect the statement, so we may suppose without loss of generality thatu(x0) =u(y0) = 0. Sinceuis lower semicontinuous, u|U being discontinuous atx0means there isε > 0and a sequence(yi)i≥1inU converging tox0such thatu(yi) ≥εfor everyi. Let(xi)be the sequence on∂U such thatyilies on the line segment joiningy0 andxi. Sinceuis convex on that segment withu(y0) = 0and u(yi) ≥ 0, we haveu(xi) ≥ u(yi) ≥ ε. Sincexi converges tox0from∂U, this shows thatu|∂Uis discontinuous atx0.
4.2. Convex envelope. Anaffine functiononRnis a function of the form a:Rn→R, a(x) =x·y+η,
whereη ∈Randx·y:=x1y1+· · ·+xnynis the standard inner product (xidenotes theithcoordinate ofx). We callythelinear partofa.
Clearly, affine functions belong toLC(Rn), hence a function given by pointwise supremum of a set of affine functions is inLC(f Rn). We can thus introduce:
Definition 4.3(Convex envelop). Given a subsetEofRn×R, theconvex envelopeof Eis the function inLC(f Rn)given by the pointwise supremum of all affine functions with epigraphs containingE. Given a setS⊂Rnand a functionϕ:S→R∪{±∞}, the convex envelope ofϕis defined as the convex envelope of the epigraph ofϕand is denoted byϕ. Equivalently,
ϕ(x) := sup{a(x)| a:Rn→Ris an affine function witha|S ≤ϕ}
This is related to the well known notion ofconvex hullof a setE⊂Rd, namely the intersection of all convex subsets ofRdcontainingE, which we denote byConv(E). We have the following important characterizations ofConv(E)and its closure:
• (See [Roc70, Theorem 17.1]) A pointx ∈Rdis inConv(E)if and only ifx is a convex combinations ofd+ 1points inE,i.e. x=t0x0+· · ·+tdxdfor somexi ∈Eandti∈[0,1]witht0+· · ·+td = 1.
• (See [Roc70, Theorem 11.5]) The closureConv(E)equals the intersection of all closed half-spaces ofRdcontainingE.
Using these facts, we can show:
Proposition 4.4. LetSbe a subset ofRnandϕ:S→R∪ {+∞}be a function.
(1) If the convex envelope ϕis not constantly−∞, then its epigraph epi(ϕ)is the closure of the convex hull ofepi(ϕ)⊂S×RinRn×R.
(2) IfSis compact andϕis lower semicontinuous, then the convex hull ofepi(ϕ)is closed, hence equalsepi(ϕ).
The assumptionϕ6≡ −∞in Part (1) means there exists an affine function ma- jorized byϕonS, which is clearly true under the assumption of Part (2). Also note that the assumption of Part (2) impliesepi(ϕ)is a closed subset ofRn+1, but the convex hull of a general unbounded closed set is not necessarily closed.
Proof. (1) Given a closed half-spaceH ⊂Rn×R∼=Rn+1, let us callH verticalif it contains a vertical line{x} ×R, and callH anupper half-spaceifH is not vertical and contains a vertical upper half-line{x} ×[ξ,+∞). Thus, upper half-spaces are exactly epigraphs of affine functions.
Ifϕ≡+∞, thenepi(ϕ)andepi(ϕ)are empty and the required conclusion holds.
Otherwise,epi(ϕ)contains a vertical upper half-line, hence every half-space con- tainingepi(ϕ)is either vertical or an upper half-space, and there is at least one up- per half-spaceH0containingepi(ϕ). For any vertical half-spaceH, the intersection H∩H0equals the intersection of all upper half-spaces containingH∩H0. There- fore, the intersection of all closed half-spaces containingepi(ϕ)coincides with the intersection of the upper half-spaces among them. This proves the required state- ment because the former intersection is the closure ofConv(epi(ϕ))and the latter isepi(ϕ)by definition ofϕ.
