Journal of Mathematical Analysis and Applications

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Contents lists available atScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

A sandwich with segment convexity

^✩

Mihály Bessenyei^∗, DávidCs. Kertész, RezsőL. Lovas

InstituteofMathematics,UniversityofDebrecen,H-4010Debrecen,Pf.12,Hungary

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

Articlehistory:

Received29September2020 Availableonline27February2021 SubmittedbyA.Daniilidis Inhonoremofourmaster,Professor JózsefSzilasi

Keywords:

Birkhoﬀsystems

Cartan–Hadamardmanifolds Convexsetsandfunctions Separationtheorems

Theaimofthisnoteistogiveasuﬃcientconditionforpairsoffunctionstohave aconvexseparatorwhentheunderlyingstructureisaCartan–Hadamardmanifold, ormoregenerally:areducedBirkhoﬀsystem.Someexoticbehaviorofconvexhulls arealsostudied.

1. Introduction

As it is well-known, separation theorems playa crucial role in many ﬁelds of Analysis and Geometry, andtheycanbe interestingontheirown right.Letusquoteheretheconvexseparationtheorem ofBaron, Matkowski,andNikodem[1],oneofourmainmotivations:

Theorem. LetD beaconvexsubsetof arealvectorspaceX,andletf,g:D→Rbegivenfunctions. There existsaconvex separatorforf andg if andonlyif

f _n

k=0

tkxk

≤ n

k=0

tkg(xk) (1)

holds for all n ∈ N, x₀,. . . ,x_n ∈ D and t₀,. . . ,t_n ∈ [0,1] with t₀+· · ·+t_n = 1. Moreover, if X is ﬁnite-dimensional,thenthelengthof theinvolvedcombinations can berestricted ton≤dim(X).

✩ SupportedbytheJánosBolyaiResearchScholarshipoftheHungarianAcademyofSciences,bytheÚNKP-20-4NewNational ExcellenceProgramsoftheMinistryforInnovationandTechnologyfromthesourceoftheNationalResearch,Developmentand InnovationFund,andbytheK-134191NKFIHGrant.

* Correspondingauthor.

E-mailaddresses:besse@science.unideb.hu(M. Bessenyei),matkdcs@uni-miskolc.hu(D.Cs. Kertész),lovas@science.unideb.hu (R.L. Lovas).

https://doi.org/10.1016/j.jmaa.2021.125108

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Thenecessitypartofthestatementisastraightforwardcalculationinbothcases.Toprovesufficiencyin thedimension-freecase,theconvexenvelopeofghastobeused.Surprisingly,themostdelicateissueissuffi- ciencyinthefinite-dimensionalsetting:AnimportanttoolofConvexGeometry,theclassicalCarathéodory Theorem [5] hasto beapplied.

The convex separation theorem above still motivates researchers. In a recent paper [13], the authors present anextension for functionsdeﬁnedoncomplete Riemannianmanifolds.Unfortunately, theirgener- alization isfalse: Asit caneasilybe seen,the two-dimensionalcasesof themain resultsof [1] and[13] do notcoincide.

The authors in [13] construct a set as the union of segments joining pairs of points of an epigraph.

Then theyclaim (withoutexplanation)itsconvexity(page 164,line7,displayedformula). Clearly,sucha construction,ingeneral,doesnotresultinaconvexset.Thustheoriginalintentof[13] remainsaniceand nontrivial challenge:Extendtheconvexseparationtheoremof[1] toRiemannianmanifolds.

In this challenge, one has to face two crucial problems. Firstly: What kind of structures should be used to haveconvexitywithoutconvex combinations? Secondly:What isthe corresponding form of(1) in lack ofalgebraic manipulations? Theproperchoice tothe structure turnsout to beBirkhoﬀ systems,the generalizationsof Cartan–Hadamardmanifolds.Inequality(1) hastobe replacedbyanotherone,inorder thataniterationprocesscanbeapplied.

2. ConvexseparationinBirkhoﬀsystems

The precise axiomatic discussion of Euclidean geometry is due to Hilbert [7]; a nice and simpliﬁed presentationcanbefoundinthebookofHartshorne[6].LaterBirkhoﬀinitiated[3] andthentogetherwith Beatley elaborated[4] an elegantanddidacticapproachwhichisbasedontherulerand theprotractor.In whatfollows,weshall needsomeoftheirnotionsandaxioms.

Assume that X is the set of points with at least two elements. Consider a family L of subsets of X whose elements aretermed lines. Letfurther d:X² → R be agiven functioncalled ametric. Werequire two axioms:Thepostulateofincidenceandthepostulateof theruler.

• Anytwodistinct pointsdetermine auniquelinecontainingthem.

• Foreach ∈L thereexistsabijectionc: R→ suchthat d

c(t),c(s)

=|t−s|.

In this case, we say that (X,L,d) is a reduced Birkhoﬀ system. A bijection c: R → satisfying the conditioninthesecondpostulateiscalledaruler for.Thepostulateoftherulerimpliesimmediatelythat thecardinality of each lineiscontinuum.

Moreover,wecanintroduceaternaryrelationcalledbetweennessonX:thepointbisbetweenthepoints a andc if a, b,c are threediﬀerent collinear points,and d(a,c)=d(a,b)+d(b,c). Usingthe abbreviation (abc) tothis fact,onecanprovethattheaxioms ofabstractbetweennessaresatisﬁed:

• If (abc),then a,b,c are pairwisedistinct andcollinear; further,(cba).

• Fordistinct pointsa,b,thereexistsc suchthat (abc).

• If (abc),then (acb)and (bac)do not hold.

Using betweenness, the notion of line segment [a,b] spanned by the points a,b can be deﬁned in the following way.Ifa=b,then[a,b]:={a}; otherwise,

[a, b] :={t∈X |(atb)} ∪ {a, b}.

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Ifa=b,andistheuniquelinepassingthroughthem,thenletc: R→bearulerforsuchthatc(α)=a andc(β)=b.Thenwecall thebijection

˜

c: [0,1]→[a, b], ˜c(t) :=c((β−α)t+α)

thestandardparametrization ofthesegment[a,b].Clearly,˜c(0)=a,and˜c(1)=b.Whenthereisnoriskof confusion,weshall alsodenote astandardparametrizationsimply byc, withouttilde. Notethat,unlike a ruler,astandardparametrization doesnotneedtobe distancepreserving (unlessd(a,b)= 1).

Once having segments, we haveconvexity concepts. A set K ⊆X is convex if [a,b] ⊆ K holds for all a,b∈K.Thefamilyofconvexsetsisdenotedby_C(X).Itturnsoutthat_C(X) isaconvexstructureindeed intheabstract senseofvandeVel[14]. Thatis,

• X and ∅are convexsets;

• theintersection of convexsetsis convex;

• theunion of nestedconvex setsisconvex.

Theconvexhull ofH ⊆X,as usual,isthesmallestconvexsetthatcontainsH: conv(H) :=

{K∈C(X)|H ⊆K}.

Itcanbeprovedthatsegmentsareconvex,andasetHisconvexifandonlyifconv(H)=H.Moreover,the mappingconv : P(X) →P(X) isahull operator,thatis, an idempotent, monotone and extensivemap.

For furtherprecise details,we refer to thepaper [2] or to theexcellent monograph [14]. Convex hullsare ﬁnitelyinner representedinthefollowingsense:

Lemma1.If (X,L,d)isareduced BirkhoﬀsystemandH ⊆X,then conv(H) =

{conv(F)|F ⊆H,card(F)<∞}.

Proof. Denote theright-handsideoftheformulaabovebyK.Then H⊆K holdsevidently;moreover,for each ﬁnite set F ⊆H, wehave thatconv(F)⊆conv(H).Thus K ⊆conv(H). To complete theproof we havetoshowthatKis convex.

If a,b ∈ K, then a ∈ conv(F_a) and b ∈ conv(F_b) with suitable ﬁnite subsets F_a and F_b of H. The set F =F_a∪F_b is ﬁnite;furthermore, conv(F_a) ⊆conv(F) and conv(F_b)⊆ conv(F) hold. Thus [a,b] ⊆ conv(F)⊆K, whichwasto beproved.

Note that convex hulls are finitely inner represented in any convex structure. The proof of this fact is based ontransfinite methods, and canbe found in the monograph [14]. When convexityis defined via segments,thepresentedelementaryapproachcanalsobe followed.

Unfortunately,neitherthedefinition,norLemma1providesaconstructivemethodforfindingtheconvex hullofaconcreteset.Especiallyfor finitesets, theformulaof Lemma1terminatesina‘circulusvitiosus’.

ThereforetheﬁxedpointtheoremofKantorovich[9] willplayadistinguishedroleforus.Itsiterationprocess isaconstructivemethodtoapproximateconvexhulls.

Lemma2.Let(X,L,d)beareduced Birkhoﬀsystem, andletH bean arbitrary subsetof X.Consider the Kantorovichiteration

H₁:=H, H_n+1 :=

{[x, y]|x, y∈H_n}.

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Then,

conv(H) =

n∈N

Hn.

Proof. Clearly,_A :={H_n|n∈N}isanincreasingchain,andH ⊆ A.Theiterationprocessguarantees that _A is a convex set. Let C ⊆ X be a convex set such that H₁ = H ⊆ C. Then H₂ ⊆ C by the convexityof C.Byinduction,Hn ⊆C foralln∈N. Thus A ⊆C.Inother words, A isthesmallest convex setcontainingH,andtheproofiscompleted.

Assumethat(X, L,d) isareducedBirkhoffsystem,andletX^∗:=X×R.Forarbitraryelements(x0,y0) and (x₁,y₁) ofX^∗, thefirstprojectionsdeterminealine in_L providedthatx₀=x₁. Letc:R→ bea rulerfor suchthatc(s₀)=x₀ andc(s₁)=x₁ hold,anddefine c^∗:R→X^∗ by

c^∗(t) =

c(at(s₁−s₀) +s₀), at(y₁−y₀) +y₀ ,

where

a:= 1

(s0−s1)²+ (y0−y1)².

Then ^∗ :={c^∗(t)|t∈R}is calledtheline connecting(x₀,y₀) and (x₁,y₁). Ifx₀=x₁, andy₀=y₁, then letc:R→X betheconstantmappinggivenbyc(t)=x0,andthendeﬁnethelineconnecting(x0,y0) and (x1,y1) bythesameformulaeas above.Inthisway, wecanspecifythelinesofX^∗ denotedby L^∗.

Finally,deﬁne themetriconX^∗by d^∗

(x₀, y₀),(x₁, y₁)

:=

d²(x₀, x₁) + (y₀−y₁)².

The triple (X^∗, L^∗,d^∗) obtained in this way will be called thevertical extension of the reduced Birkhoﬀ system(X,L,d).Themostimportantpropertyofverticalextensionsissubsumedbythefollowinglemma.

Lemma 3.The verticalextension ofareduced Birkhoﬀsystemis areducedBirkhoﬀ system.

Proof. It caneasily be checkedthat thepostulate of incidenceis valid in thevertical extension. Keeping thepreviousnotations,consideraline^∗determinedby(x0,y0) and(x1,y1) withdistinctﬁrstprojections.

Weclaimthatc^∗ servesasarulerfor^∗.Indeed,sincecisarulerfor,wearriveat d^∗(c^∗(t), c^∗(s)) =

a²(t−s)²(s₁−s₀)²+a²(t−s)²(y₁−y₀)²

=|t−s|a

(s₀−s₁)²+ (y₀−y₁)²=|t−s|. If x0=x1,thecaseofverticallines, canbe handledsimilarly.

Assume that D is a nonempty convex subset in a reduced Birkhoﬀ system. We say that a function ϕ:D→R issegmentconvex,or simply:convex,if

ϕ(c(t))≤(1−t)ϕ(x₀) +tϕ(x₁)

holds for all x₀,x₁ ∈ D and for all t ∈ [0,1], where c: [0,1] → D is the unique line determined by the propertiesc(0)=x₀ andc(1)=x₁.

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Ourmain resultgivesasuﬃcient conditionfortheexistenceof aconvexseparator. Toformulateit, we need thefollowing concept.We saythatareduced Birkhoﬀsystem (X,_L,d) is dropcomplete if,for each convexset K⊆X andforallx0∈X,theusualdrop representationholds:

conv({x₀} ∪K) =

{[x₀, x]|x∈K}.

Nowwe canformulateourmainresult, theadequateversionofthe Baron–Matkowski–Nikodemconvex separationtheorem. Wepoint out thatit isnot adirectcopy of theclassical result:contrary to (1),both functionshaveto beinvolvedintheupperestimation.Themixedformenablesustoiterateourinequality recursively.

Theorem 1.LetD be aconvex set ina reduced Birkhoﬀsystem (X,_L,d) whose vertical extension isdrop complete. If, for all n ∈ N, x₀,. . . ,x_n ∈ D, x ∈ conv{x₁,. . . ,x_n}, and for all t ∈ [0,1], the functions f,g:D→Rsatisfy theinequality

f c(t)

≤(1−t)g(x₀) +tf(x), (2)

where c: [0,1] → D is the segment joining x0 = c(0) and x = c(1), then there exists a convex function ϕ:D→R fulﬁllingf ≤ϕ≤g.

Proof. Let E:= conv(epi(g)).Firstwe showthatf(x)≤y whenever(x,y)∈E. ByLemma1,eachpoint ofE belongstotheconvexhullofsomeﬁnitesubsetofepi(g).If(x,y) belongstoasingleton,thenf(x)≤y holdstrivially.Assumethatthedesiredinequalityholdsif(x,y) belongstotheconvexhullofanynelement subsetofepi(g).Considerthecasewhen

(x, y)∈conv{(x₀, y₀), . . . ,(x_n, y_n)} and g(x₀)≤y₀, . . . , g(x_n)≤y_n.

The vertical extension is adrop complete reduced Birkhoﬀ system, thusthere exists apoint (x^∗,y^∗) and t∈[0,1] suchthat

(x^∗, y^∗)∈conv{(x1, y1), . . . ,(xn, yn)} and (x, y) =

c(t),(1−t)y0+ty^∗ ,

where c: [0,1]→D isthesegment fulﬁllingc(0)=x0 and c(1)=x^∗. Usingtheinductive assumptionand (2),wearriveat

f(x) =f c(t)

≤(1−t)g(x0) +tf(x^∗)≤(1−t)y0+ty^∗=y, whichwasourclaim.Thispropertyensuresthattheformula

ϕ(x) := inf{y∈R|(x, y)∈E}

deﬁnes a functionϕ:D →R. Clearly, f ≤ ϕ≤ g. Finally we prove thatϕ is convex. Let x0,x1 ∈D be arbitraryandchoosey0,y1∈Rsuchthat(x0,y0) and (x1,y1) belongtoE. SinceE isconvex,

c(t),(1−t)y0+ty1

∈E

holdsforallt∈[0,1],wherec: [0,1]→Disthesegmentfulﬁllingc(0)=x0andc(1)=x1.Bythedeﬁnition ofϕ, wehavethatϕ

c(t)

≤(1−t)y₀+ty₁.Taking theinﬁmumat y₀ and y₁,we gettheconvexityof ϕ, andthiscompletes theproof.

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Assume thattheunderlyingreduced Birkhoﬀ systemis avectorspace. Then, using induction,one can checkeasilythat(2) implies(1).Unfortunately,theconverseimplicationisnotvalid.Thusourmainresults (thepreviousandthenexttheorems)areonlysuﬃcientconditionsfortheexistenceofaconvexseparator.

However,underthisgenerality,afullcharacterizationcannotbeexpected.

ConsiderareducedBirkhoﬀ system(X,L,d),andlet

K:={k∈N| ∀H ⊆X,∀x∈conv(H) :∃F ⊆H :x∈conv(F),card(F)≤k}.

If K isnot empty,then κ:= minK is called theCarathéodory number of thesystem; and ifK is empty, thentheCarathéodorynumberisdefinedtobe+∞.Equivalently,theCarathéodorynumbercanbedefined as the leastκ suchthattherepresentation of Lemma1remainsvalid ifwe allowonly theconvex hullsof sets withat mostκelementson theright-handside,or as+∞ifthere isnofinite κwiththisproperty. If theCarathédorynumberoftheverticalextensionisknown,wecanstrengthenthestatementofTheorem1.

Theproof isessentiallythesame,thusweomitit.

Theorem 2.Keeping the conditions of the previous theorem, assume that the Carathéodory number of the vertical extension isκ.Thenthesize oftheinvolved convexhullcan bereduced ton≤κ.

If X is a ﬁnite-dimensional vector space, the classical separation result of [1] restricts the length of the involved combination to dim(X)+ 1, while the Carthéodory numberof thevertical extension is κ= dim(X)+ 2.Inotherwords,thereductionofTheorem2canbeimprovedintheﬁnite-dimensionalsetting.

ThedropcompletenessoftheverticalextensionsisrequiredbothinTheorem1andTheorem2.Clearly, this assumption makes theunderlying reduced Birkhoﬀ systemdrop complete as well.Thus thequestion arises, quiteevidently:Does theextension inherit dropcompletenessfrom theoriginal system? Inorder to justify theconditionsof themain results,wewillgiveanegative answerinthelastsection.

3. ConvexseparationinCartan–Hadamardmanifolds

The celebrated theorem of Hopf states that each simply connected, complete Riemannian manifold of positive sectional curvature is compact. In contrast to this behavior, nonpositive curvature results in an oppositefeatureaccordingto thetheorem ofCartanandHadamard:

Theorem. The exponentialmapatanypointofasimplyconnected,complete Riemannianmanifoldofnon- positive sectionalcurvature isaglobal diﬀeomorphismbetween thetangent spaceatthegivenpointand the manifold.

These manifolds are called Cartan–Hadamard manifolds. In particular, by this theorem, each Cartan–

HadamardmanifoldishomeomorphictoaEuclideanspace.Moreover,geodesicscanbeparametrizedalong theentiresetofreals,andtwogeodesicscanhaveatmostonecommonpoint.Thismeansthatthepostulate of incidenceandthepostulateofruleraresatisﬁed,andwecanformulatethenextstatement.

Lemma 4.Each Cartan–Hadamardmanifoldisareduced Birkhoﬀsystem.

ByLemma3andLemma4,theverticalextensionofaCartan–HadamardmanifoldisareducedBirkhoﬀ system.Moreover,nowtheverticalextensionhasaverycloserelationwiththeproductRiemannianmetric.

Infact,exactlythisrelation(formulatedinthenextlemma)hasmotivatedthenotionofverticalextensions.

Forthetechnicalbackgroundoftheproof,we referto themonograph ofSakai[12].

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Lemma 5.If M isa Cartan–Hadamard manifold, then M×R is also a Cartan–Hadamardmanifold with respecttothe productRiemannianstructure,andtheinduced Birkhoﬀstructure coincideswith thevertical extensionof theBirkhoﬀstructure ofM.

Proof. Let d be the dimension of M, and denote the components of the metric tensor by gij. Then the metrictensoranditsinverseoftheproductmanifoldM×R arerepresentedas

(Gij) =

(gij) 0

0 1

and (G^ij) =

(g^ij) 0

0 1

.

Clearly, theproduct manifold is asimplyconnected and complete Riemannianmanifold. Moreover,using theKoszulformulae

Γ^k_ij = 1 2G^kl

∂Gjl

∂xⁱ +∂Gli

∂x^j −∂Gij

∂x^l

forthe Christoﬀelsymbols ofM×R, wecanconcludethatΓ^k_ij = 0 whenever(d+ 1)∈ {i,j,k}. Consider now ageodesicc^∗ intheproduct manifold M×R. Sinceits coordinatefunctionssatisfy thesecond-order diﬀerentialequations

c^∗k+

Γ^k_ij◦c^∗

c^∗ic^∗j = 0

and theChristoffel symbols havethepreviously mentionedproperties, c^∗^(d+1) = 0 follows. Thus thelast component of c^∗ is affine. Furthermore, by the behavior of the Christoffel symbols and by the geodesic differential equationagain, thefirst projectionof c^∗ resultsinageodesicof M.Therefore any geodesicc^∗ connectingthepoints(x0,y0) and(x1,y1) oftheproduct manifoldM×R canbe parametrizedas

c^∗(t) =

c(t),(1−t)y0+ty1

,

wherethegeodesiccofM connectsthepointsc(0)=x0 andc(1)=x1.Notealsothatthisparametrization isaglobaloneincaseofCartan–Hadamardmanifolds.ThisshowsthattheproductmanifoldM×Risthe verticalextension ofM.

Inparticular,theverticalextensionisasimplyconnectedandcomplete manifold,aswell.Nowweshow thatitssectionalcurvature isnonpositive. Letσ⊂T_(x,y)(M×R)=T_xM⊕Rbe anarbitrary plane,and let b1,b2 be a base in σ. If TxM∩σ = {0}, then dim(T(x,y)(M×R)) =d+ 2, which is a contradiction.

Thusdim(TxM∩σ)≥1,andwemayassumethatthesecond directcomponentofb2 iszero.Thenb1 has adirectdecompositionb1=b11+b12.Recall thatthesignof thesectionalcurvature dependsonly onthe signof R(b1,b2,b2,b1),where RistheRiemanniancurvaturetensor.Sinceitscomponentscanbeobtained by

Rijkl= ∂Γ^m_jk

∂xⁱ −∂Γ^m_ik

∂x^j + Γ^r_jkΓ^m_ir−Γ^r_ikΓ^m_jr

Glm,

wearriveatR_ijkl= 0 providedthat(d+ 1)∈ {i,j,k,l}.Thus_R vanishesifoneoftheargumentscontains b₁₂.Therefore,

R(b1, b2, b2, b1) =R(b11+b12, b2, b2, b11+b12) = R(b11, b2, b2, b11) +R(b12, b2, b2, b11)+

R(b11, b2, b2, b12) +R(b12, b2, b2, b12) =R(b11, b2, b2, b11)≤0,

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sincethesecond directcomponentsofb₁₁ andb₂ arezeroandthesectionalcurvatureof M isnonpositive.

This completestheproof.

For moredetailsonCartan–Hadamardmanifolds, we recommendthebook ofJost[8].As directconse- quencesofTheorem1andTheorem2,wecanformulatethenexttwocorollaries.

Corollary 1.Let D be a convex set in a Cartan–Hadamard manifold M whose vertical extension is drop complete. If, for all n ∈ N, x0,. . . ,xn ∈ D, x ∈ conv{x1,. . . ,xn}, and for all t ∈ [0,1], the functions f,g:D → R satisfy (2) where c: [0,1]→ R is the geodesic segment joining x0 =c(0) and x=c(1), then there existsaconvexfunctionϕ:D→R fulﬁllingf ≤ϕ≤g.

Corollary 2.Keeping theconditions of the previoustheorem, assume that the Carathéodory number of the vertical extension isκ.Thenthesize oftheinvolved convexhullcan bereduced ton≤κ.

Cartan–HadamardmanifoldsandEuclideanspacesarequite“close”relatives.Henceonemayexpectthat the Carathéodory numberofa Cartan–Hadamardmanifold M, accordinglyto theEuclidean case, can be expressed asκ= dim(M)+ 1.However,as faras weknow,thisisstillanopenproblemposedbyLedyaev, Treiman,andZhu [10].

4. Theexoticbehaviorof convexhulls

The aimof this sectionis to provethatdropcompleteness ofthe vertical extensioncannot be changed todropcompletenessoftheunderlyingsysteminTheorem1andTheorem2.Themainreasonistheexotic behaviorofconvexhulls:ItmayoccurthattheconvexhullofthreepointsinaCartan–Hadamardmanifold is notcontained inatwo-dimensionalsubmanifold. Toconstruct suchanexample, letus recallheresome basicfactsinhyperbolicgeometry. Forreferences,seethebookofRatcliﬀe[11].

Thehyperbolicplane,denotedbyH² intheforthcomings,isatwo-dimensionalCartan–Hadamardman- ifold with constant sectional curvature −1. We will use two of its several models. The ﬁrst one is the Beltrami–Klein model(known alsoas theCayley–Klein model):Here theplane istheopen unitdisc, and thelines areitsEuclideanchordsegments.Thedistance ofthismodelwillnotbeused.

The second modelis the Poincaréhalf-plane model.The plane is the upper open Cartesianhalf-plane R×R₊;linesareeithercircleswithcenterontheboundaryline orverticalEuclideanhalf-lines.Itsmetric plays akeyrole inourinvestigation.Thedistanceofa= (a₁,a₂) andb= (b₁,b₂) isgivenby

d(a, b) = 2 ln

(a1−b1)²+ (a2−b2)²+

(a1−b1)²+ (a2+b2)² 2√

a₂b₂ . (3)

Inparticular, ifa1=b1,this formulacanbe simpliﬁed(whichwillbe quiteconvenientforus):

d(a, b) =|lna₂−lnb₂|. (4)

If(x₀,y₀) and(x₁,y₁) arepointsoftheverticalextensionH²×R,andc: [0,1]→H²istheuniquegeodesic fulﬁlling c(0)=x0 and c(1) =x1, then the unique geodesicsegment c^∗: [0,1]→ H²×R which connects (x0,y0) and(x1,y1) isgivenby

c^∗(t) =

c(t),(1−t)y₀+ty₁ .

Since geodesicsare ofconstantspeed,wehaved(x₀,x)=td(x₀,x₁) forall t∈[0,1],where x=c(t). Thus we canreconstructthepoints (x,y) between(x₀,y₀) and(x₁,y₁) fromxas

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(x, y) =

x, y0+ d(x₀, x)

d(x0, x1)(y1−y0)

. (5)

Theorem3.The hyperbolic planeis dropcomplete,whileitsvertical extensionis not.

Proof. TheBeltrami–KleinmodelshowsthattheconvexstructureofH²canbe identiﬁedwiththeconvex structureof theopen discinheritedfrom R².Inparticular,thedroprepresentationisvalidinH².

Now we provethat thevertical extension H²×R is notdrop complete. Wewill illustrate it using the convexhullofthepoints

A= ((0,3),0); B= ((4,5),1); C= ((−4,5),1).

Considertheirprojectionsa,b,c andtheadditionalpointsp,q,ronH²:

a= (0,3), b= (4,5), c= (−4,5); p= (1,4), q= (−1,4), r= (0,√ 41).

Itisimmediatetocheck thatp∈[a,b] andq∈[a,c],furthermorer∈[b,c] (thesegmentsherearemeantin thehyperbolic geodesicsense:theyarearcsof circles).Finally,weneed theintersectionof[p,q] and[a,r], whichturnsouttobe x= (0,√

17).Thenextﬁgureshowsthese choices:

LetP and Qbe the pointson [A,B] and [A,C] in thevertical extension,whose ﬁrst projectionsare p andq,respectively.Nowwereconstructtheirlastcoordinatesfromtheprojections.Bythedistanceformula (3),

d(a, b) = 2 ln

√20 +√ 80 2√

15 = ln 3 and d(a, p) = 2 ln

√2 +√ 50 2√

12 = ln 3−ln 2.

ThusthesecondcommonprojectionofP andQisobtainedvia(5) as ε1:= d(a, p)

d(a, b) = 1−ln 2 ln 3 <2

5.

Theestimation abovecanbechecked evenby hand.Moreover,thepoints ofthesegment [P,Q] share this commonlastcomponent.Therefore,

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X1:=

(0,√ 17), ε1

∈[P, Q]⊆conv{A, B, C}.

Clearly, R = ((0,√

41),1) ∈ [B,C]. Now we determine the point of the vertical extension whose ﬁrst projectionisxandbelongstothesegment[A,R].Using(4) and(5),itslastcomponentturnsouttobe

ε2:=ln√

17−ln 3 ln√

41−ln 3 = ln 17−2 ln 3 ln 41−2 ln 3 >2

5. This estimation,withabitmoreeﬀort,canalsobe checkedbyhand.Thus,

X₂:=

(0,√ 17), ε₂

∈[A, R]⊆

{[A, D]|D∈[B, C]}.

Since X1=X2, wecanconclude thatthedrop representationinvolving {A}and [B,C] does notcover theentireconvex hullofA,B,C,whichwastobe proved.

TheBeltrami–Kleinopenspheremodelandtheﬁrstpartoftheargumentshowthatthegeodesicconvex structureofhyperbolicspaceiscompatiblewiththeEuclideanconvexstructure oftheopenball.Inparticular, thehyperbolic spaceisdropcompleteinanydimension,anditscombinatorialinvariantscoincidewiththe standardEuclidean ones.UsingtheCartan–Hadamard theoremor theresultsof[2], itcanbeprovedthat these propertiesarealsotruefortwo-dimensionalCartan–Hadamard manifolds.

As we have already mentioned, the greatest advantage of Lemma2 is that it can be implemented. In fact, the theorem abovewas conjectured viaacomputer algorithm.In whatfollows, we sketchbrieﬂy its pseudo code.

Weshallneedtwo functions.Theﬁrstone,geodcalculates apoint ofageodesicbetweentwopoints:

geod: (H²×R)×(H²×R)×R→H²×R

sothatgeod(A,B,0)=Aandgeod(A,B,1)=Bhold.Thefunctiondistcalculatesthehyperbolicdistance of twopointsinH²:

dist(a, b) :H²×H²→R.

ThelistConvexHullcollectsthepointsoftheconvexhullasanorderedlist.Initiallyweputthepointsofthe setwhoseconvexhullistobecomputedintoConvexHull.ConvexHull[i] istheithelementinConvexHull.

Indexingstartswith0.Theparameteriterationsisthenumberofiterations.Finally,theparameterres is thehyperbolic distancebetweenpointsto becalculatedalonggeodesics.

ThealgorithmtakestwopointsAandBfromConvexHullandaddsthepointsofthegeodesicfromAto B withhyperbolic distanceresfromeachother toConvexHull.ThepointB ischosenso thatrepetitions are avoided.

• Thevariablessizeandprevioussizekeeptrackofthenumberofpointscalculatedintheconvexhull.

Initially

– size:=|ConvexHull|. – previoussize:= 0.

• Thefollowingloopistobe performediterationsmanytimes.

– Fori= 0,. . . ,size−2,performthefollowing:

∗ Forj= max(previoussize,i+ 1),. . . ,size−1,performthefollowing:

· A:=ConvexHull[i].

· B :=ConvexHull[j].

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· d:=dist(A,B).

· Fork= 1,2,. . . whilek·res< d,

addgeod(A,B,k·res/d) toConvexHull.

– previoussize:=size;

– size:=|ConvexHull|.

Usingouralgorithm,wecanillustratesomepointsofconv{A,B,C}intheproofofTheorem3.Theﬁrst ﬁgureistheintersectionoftheapproximationoftheconvexhullwiththeplane[a,r]×R:

Thesecondﬁgureistheintersectionof theapproximationwith [p,q]×R:

Ineachcase,iterations= 2 andres= 0.006.Intheconcreteimplementation,pointsareplottedwhose distancefrom theplanesisatmost0.01.

Theprogramis availableandfreelydownloadablefrom thehomepagebelow:

http://shrek.unideb.hu/~ftzydk/convex/

Acknowledgment

We wish to express our gratitude to professor Sándor Kristály for the valuable discussions on this topic.

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