A sandwich with segment convexity
AA SANDWICH WITH SEGMENT CONVEXITY
MIHÁLY BESSENYEI, DÁVID CS. KERTÉSZ, AND REZS ˝O L. LOVAS
In honorem of our master, Professor József Szilasi A BSTRACT . The aim of this note is to give a sufficient condition for pairs of functions to have aconvex separator when the underlying structure is a Cartan–Hadamard manifold, or more generally:a reduced Birkhoff–Beatley system. Some exotic behavior of convex hulls are also studied.
1. I
NTRODUCTION
As it is well-known, separation theorems play a crucial role in many fields of Analysis andGeometry, and they can be interesting on their own right. Let us quote here the convex separationtheorem of Baron, Matkowski, and Nikodem [1], one of our main motivations:
Theorem.
Let D be a convex subset of a real vector space X , and let f, g : D → R be givenfunctions. There exists a convex separator for f and g if and only if (1) f (cid:32) n (cid:88) k =0 t k x k (cid:33) ≤ n (cid:88) k =0 t k g ( x k ) holds for all n ∈ N , x , . . . , x n ∈ D and t , . . . , t n ∈ [0 , with t + · · · + t n = 1 . Moreover, if X isfinite dimensional, then the length of the involved combinations can be restricted to n ≤ dim( X ) . The necessity part of the statement is a straightforward calculation in both cases. To provesufficiency in the dimension-free case, the convex envelope of g has to be used. Surprisingly, themost delicate issue is sufficiency in the finite dimensional setting: An important tool of ConvexGeometry, the classical Carathéodory Theorem [5] has to be applied.The convex separation theorem above still motivates researchers. In a recent paper [13], theauthors present an extension for functions defined on complete Riemannian manifolds. Unfortu-nately, their generalization is false: As it can easily be seen, the two dimensional cases of the mainresults of [1] and [13] do not coincide.The authors in [13] construct a set as the union of segments joining pairs of points of an epi-graph. Then they claim (without explanation) its convexity (page , line , displaced formula).Clearly, such a construction, in general, does not result in a convex set. Thus the original intentof [13] remains a nice and nontrivial challenge: Extend the convex separation theorem of [1] toRiemannian manifolds. Date : September 29, 2020.2010
Mathematics Subject Classification.
Primary 26B25; Secondary 39B62, 39B82, 52A05, 52A55, 53C22,53C25.
Key words and phrases.
Birkhoff–Beatley systems, Cartan–Hadamard manifolds, convex sets and functions, sepa-ration theorems.This paper was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by theÚNKP-19-4 New National Excellence Programs of the Ministry for Innovation and Technology, and by the K-134191NKFIH Grant. a r X i v : . [ m a t h . M G ] S e p M. BESSENYEI, D. CS. KERTÉSZ, AND R. L. LOVAS
In this challenge, one has to face two crucial problems. Firstly: What kind of structures shouldbe used to have convexity without convex combinations? Secondly: What is the correspondingform of (1) in lack of algebraic manipulations? The proper choice to the structure turns out to beBirkhoff–Beatley systems, the generalizations of Cartan–Hadamard manifolds. Inequality (1) hasto be replaced by another one, in order that an iteration process can be applied.2. C
ONVEX SEPARATION IN B IRKHOFF –B EATLEY SYSTEMS
The precise axiomatic discussion of Euclidean geometry is due to Hilbert [7]; a nice and simpli-fied presentation can be found in the book of Hartshorne [6]. Later Birkhoff initiated [3] and thentogether with Beatley elaborated [4] an elegant and didactic approach which is based on the rulerand the protractor. In what follows, we shall need some of their notions and axioms.Assume that X is the set of points with at least two elements. Consider a family L of subsetsof X whose elements are termed lines . Let further d : X → R be a given function called a metric .We require two axioms: The postulate of incidence and the postulate of the ruler: • Any two distinct points determine a unique line containing them. • For each (cid:96) ∈ L there exists a bijection c : R → (cid:96) such that d (cid:0) c ( t ) , c ( s ) (cid:1) = | t − s | . In this case, we say that ( X, L , d ) is a reduced Birkhoff–Beatley system . A bijection c : R → (cid:96) satisfying the condition in the second postulate is called a ruler for (cid:96) .The postulate of the ruler implies immediately, that each line has at least two points . Moreover,we can introduce a ternary relation called betweenness on X : the point b is between the points a and c if a , b , c are three different collinear points, and d ( a, c ) = d ( a, b ) + d ( b, c ) . Using theabbreviation ( abc ) to this fact, one can prove that the axioms of abstract betweenness are satisfied: • If ( abc ) , then a, b, c are pairwise distinct and collinear; further, ( cba ) . • For distinct points a, b , there exists c such that ( 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 inthe following way. If a = b , then [ a, b ] := { a } ; otherwise, [ a, b ] := { t ∈ X | ( atb ) } ∪ { a, b } . If a (cid:54) = b , and (cid:96) is the unique line passing through them, then let c : R → (cid:96) be a ruler for (cid:96) such that c ( α ) = a and c ( β ) = b . Then we call the bijection ˜ c : [0 , → [ a, b ] , ˜ c ( t ) := c (( β − α ) t + α ) the standard parametrization of the segment [ a, b ] . Clearly, ˜ c (0) = a , and ˜ c (1) = b . When thereis no risk of confusion, we shall also denote a standard parametrization simply by c , without tilde.Note that, unlike a ruler, a standard parametrization does not need to be distance preserving (unless d ( a, b ) = 1 ).Once having segments, we have convexity concepts. A set K ⊆ X is convex if [ a, b ] ⊆ K holdsfor all a, b ∈ K . The family of convex sets is denoted by C ( X ) . It turns out that C ( X ) is a convexstructure indeed in the abstract sense of van de Vel [14]. That is, • X and ∅ are convex sets; • the intersection of convex sets is convex; • the union of nested convex sets is convex. SANDWICH WITH SEGMENT CONVEXITY 3
The convex hull of H ⊆ X , as usual, is the smallest convex set that contains H :conv ( H ) := (cid:92) { K ∈ C ( X ) | H ⊆ K } . It can be proved that segments are convex, and a set H is convex if and only if conv ( H ) = H .Moreover, the mapping conv : P ( X ) → P ( X ) is a hull operator, that is, an idempotent, mono-tone and extensive map. For further precise details, we refer to the paper [2] or to the excellentmonograph [14]. Convex hulls are finitely inner represented in the following sense: Lemma 1. If ( X, L , d ) is a reduced Birkhoff–Beatley system and H ⊆ X , then conv ( H ) = (cid:91) { conv ( F ) | F ⊆ H, card ( F ) < ∞} . Proof.
Denote the right-hand side of the formula above by K . Then H ⊆ K holds evidently;moreover, for each finite set F ⊆ H , we have that conv ( F ) ⊆ conv ( H ) . Thus K ⊆ conv ( H ) . Tocomplete the proof we have to show that K is 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 , which was to be proved. (cid:3) Note, that convex hulls are finitely inner represented in any convex structure. The proof of thisfact is based on transfinite methods, and can be found in the monograph [14]. When convexity isdefined via segments, the presented elementary approach can also be followed.Unfortunately, neither the definition, nor Lemma 1 provides a constructive method for findingthe convex hull of a concrete set. Especially for finite sets, the formula of Lemma 1 terminates in a‘circulus vitiosus’. Therefore the fixed point theorem of Kantorovich [9] will play a distinguishedrole for us. Its iteration process is a constructive method to approximate convex hulls.
Lemma 2.
Let ( X, L , d ) be a reduced Birkhoff–Beatley system, and let H be an arbitrary subsetof X . Consider the Kantorovich iteration H := H, H n +1 := (cid:91) { [ x, y ] | x, y ∈ H n } . Then, conv ( H ) = (cid:91) n ∈ N H n . Proof.
Clearly, L := { H n | n ∈ N } is an increasing chain, and H ⊆ (cid:83) L . The iteration processguarantees that (cid:83) L is a convex set. Let C ⊆ X be a convex set such that H = H ⊆ C . Then H ⊆ C by the convexity of C . By induction, H n ⊆ C for all n ∈ N . Thus (cid:83) L ⊆ C . In otherwords, (cid:83) L is the smallest convex set containing H , and the proof is completed. (cid:3) Assume that ( X, L , d ) is a reduced Birkhoff–Beatley system, and let X ∗ := X × R . Forarbitrary elements ( x , y ) and ( x , y ) of X ∗ , the first projections determine a line (cid:96) in L providedthat x (cid:54) = x . Let c : R → (cid:96) be a ruler for (cid:96) such that c ( s ) = x and c ( s ) = x hold, and define c ∗ : R → X ∗ by c ∗ ( t ) = (cid:0) c ( at ( s − s ) + s ) , at ( y − y ) + y (cid:1) , where a := 1 (cid:112) ( s − s ) + ( y − y ) . M. BESSENYEI, D. CS. KERTÉSZ, AND R. L. LOVAS
Then (cid:96) ∗ := { c ∗ ( t ) | t ∈ R } is called the line connecting ( x , y ) and ( x , y ) . If x = x , and y (cid:54) = y , then let c : R → X be the constant mapping given by c ( t ) = x , and then define the lineconnecting ( x , y ) and ( x , y ) by the same formulae as above. In this way, we can specify the lines of X ∗ denoted by L ∗ .Finally, define the metric on X ∗ by d ∗ (cid:0) ( x , y ) , ( x , y ) (cid:1) := (cid:112) d ( x , x ) + ( y − y ) . The triple ( X ∗ , L ∗ , d ∗ ) obtained in this way will be called the vertical extension of the reducedBirkhoff–Beatley system ( X, L , d ) . The most important property of vertical extensions is sub-sumed by the following lemma. Lemma 3.
The vertical extension of a reduced Birkhoff–Beatley system is a reduced Birkhoff–Beatley system.Proof.
It can easily be checked that the postulate of incidence is valid in the vertical extension.Keeping the previous notations, consider a line (cid:96) ∗ determined by ( x , y ) and ( x , y ) with distinctfirst projections. We claim that c ∗ serves as a ruler for (cid:96) ∗ . Indeed, since c is a ruler for (cid:96) , we arriveat d ∗ ( c ∗ ( t ) , c ∗ ( s )) = (cid:112) a ( t − s ) ( s − s ) + a ( t − s ) ( y − y ) = | t − s | a (cid:112) ( s − s ) + ( y − y ) = | t − s | . If x = x , the case of vertical lines, can be handled similarly. (cid:3) Assume that D is a nonempty convex subset in a reduced Birkhoff–Beatley system. We say thata function ϕ : D → R is segment convex , or simply: convex , if ϕ ( c ( t )) ≤ (1 − t ) ϕ ( x ) + tϕ ( x ) holds for all x , x ∈ D and for all t ∈ [0 , , where c : [0 , → D is the unique line determinedby the properties c (0) = x and c (1) = x .Our main result gives a sufficient condition for the existence of a convex separator. To formulateit, we need the following concept. We say that a reduced Birkhoff–Beatley system ( X, L , d ) is drop complete if, for each convex set K ⊆ X and for all x ∈ X , the usual drop representationholds: conv ( { x } ∪ K ) = (cid:91) { [ x , x ] | x ∈ K } . Theorem 1.
Let D be a convex set in a reduced Birkhoff–Beatley system ( X, L , d ) whose verticalextension is drop complete. If, for all n ∈ N , x , . . . , x n ∈ D , x ∈ conv { x , . . . , x n } , and for all t ∈ [0 , , the functions f, g : D → R satisfy the inequality (2) f (cid:0) c ( t ) (cid:1) ≤ (1 − t ) g ( x ) + tf ( x ) , where c : [0 , → D is the segment joining x = c (0) and x = c (1) , then there exists a convexfunction ϕ : D → R fulfilling f ≤ ϕ ≤ g .Proof. Let E := conv ( epi ( g )) . First we show, that f ( x ) ≤ y whenever ( x, y ) ∈ E . By Lemma 1,each point of E belongs to the convex hull of some finite subset of epi ( g ) . If ( x, y ) belongs to asingleton, then f ( x ) ≤ y holds trivially. Assume that the desired inequality holds if ( x, y ) belongsto the convex hull of any n element subset of epi ( g ) . Consider the case when ( x, y ) ∈ conv { ( x , y ) , . . . , ( x n , y n ) } and g ( x ) ≤ y , . . . , g ( x n ) ≤ y n . SANDWICH WITH SEGMENT CONVEXITY 5
The vertical extension is a drop complete reduced Birkhoff–Beatley system, thus there exists apoint ( x ∗ , y ∗ ) and t ∈ [0 , such that ( x ∗ , y ∗ ) ∈ conv { ( x , y ) , . . . , ( x n , y n ) } and ( x, y ) = (cid:0) c ( t ) , (1 − t ) y + ty ∗ (cid:1) , where c : [0 , → D is the segment fulfilling c (0) = x and c (1) = x ∗ . Using the inductiveassumption and (2), we arrive at f ( x ) = f (cid:0) c ( t ) (cid:1) ≤ (1 − t ) g ( x ) + tf ( x ∗ ) ≤ (1 − t ) y + ty ∗ = y, which was our claim. This property ensures that the formula ϕ ( x ) := inf { y ∈ R | ( x, y ) ∈ E } defines a function ϕ : D → R . Clearly, f ≤ ϕ ≤ g . Finally we prove that ϕ is convex. Let x , x ∈ D be arbitrary and choose y , y ∈ R such that ( x , y ) and ( x , y ) belong to E . Since E is convex, (cid:0) c ( t ) , (1 − t ) y + ty (cid:1) ∈ E holds for all t ∈ [0 , , where c : [0 , → D is the segment fulfilling c (0) = x and c (1) = x . Bythe definition of ϕ , we have that ϕ (cid:0) c ( t ) (cid:1) ≤ (1 − t ) y + ty . Taking the infimum at y and y , weget the convexity of ϕ , and this completes the proof. (cid:3) Let H be an arbitrary subset of a reduced Birkhoff–Beatly system, and assume that for each x ∈ conv ( H ) there exists F ⊆ H such that x ∈ conv ( F ) and card ( F ) ≤ κ . The least possible κ with this property is called the Carathéodory number of the system. If there does not existsuch a κ , then the Carathéodory number is defined to be + ∞ . Equivalently, the representation ofLemma 1 remains valid if we allow only the convex hulls of sets with at most κ elements on theright-hand side. If the Carathédory number of the vertical extension is known, we can strengthenthe statement of Theorem 1. The proof is essentially the same, thus we omit it. Theorem 2.
Keeping the conditions of the previous theorem, assume that the Carathéodory num-ber of the vertical extension is κ . Then the size of the involved convex hull can be reduced to n ≤ κ . Assume that the underlying reduced Birkhoff–Beatly system is a vector space. Then, usinginduction, one can check easily that (2) implies (1). Unfortunately, the converse implication is notvalid. Thus our main results are only sufficient conditions for the existence of a convex separator.However, under this generality, a full characterization cannot be expected.If X is a finite dimensional vector space, the classical separation result of [1] restricts the lengthof the involved combination to dim( X )+1 , while the Carthéodory number of the vertical extensionis κ = dim( X ) + 2 . In other words, the reduction of Theorem 2 can be improved in the finitedimensional setting.The drop completeness of the vertical extensions is required both in Theorem 1 and Theorem 2.Clearly, this assumption makes the underlying reduced Birkhoff–Beatley system drop complete aswell. Thus the question arises, quite evidently: Does the extension inherit drop completeness fromthe original system?
In order to justify the conditions of the main results, we will give a negativeanswer in the last section.
M. BESSENYEI, D. CS. KERTÉSZ, AND R. L. LOVAS
3. C
ONVEX SEPARATION IN C ARTAN –H ADAMARD MANIFOLDS
The celebrated theorem of Hopf states that each simply connected, complete Riemannian mani-fold of positive sectional curvature is compact.
In contrast to this behavior, nonpositive curvatureresults in an opposite feature according to the theorem of Cartan and Hadamard:
Theorem.
The exponential map at any point of a simply connected, complete d dimensional Rie-mannian manifold of nonpositive sectional curvature is a global diffeomorphism between R d andthe manifold. These manifolds are called
Cartan–Hadamard manifolds . In particular, by this theorem, eachCartan–Hadamard manifold is homeomorphic to a Euclidean space. Moreover, geodesics can beparametrized along the entire set of reals, and two geodesics can have at most one common point.This means that the postulate of incidence and the postulate of ruler are satisfied, and we canformulate the next statement.
Lemma 4.
Each Cartan–Hadamard manifold is a reduced Birkhoff–Beatley system.
By Lemma 3 and Lemma 4, the vertical extension of a Cartan–Hadamard manifold is a reducedBirkhoff–Beatley system. Moreover, now the vertical extension has a very close relation withthe product Riemannian metric. In fact, exactly this relation (formulated in the next lemma) hasmotivated the notion of vertical extensions. For the technical background of the proof, we refer tothe monograph of Sakai [12].
Lemma 5. If M is a Cartan–Hadamard manifold, then M × R is also a Cartan–Hadamard mani-fold with respect to the product Riemannian structure, and the induced Birkhoff–Beatley structurecoincides with the vertical extension of the Birkhoff–Beatley structure of M .Proof. Let d be the dimension of M , and denote the components of the metric tensor by g ij . Thenthe metric tensor and its inverse of the product manifold M × R are represented as ( G ij ) = (cid:18) ( g ij ) 00 1 (cid:19) and ( G ij ) = (cid:18) ( g ij ) 00 1 (cid:19) . Clearly, the product manifold is a simply connected and complete Riemannian manifold. More-over, using the Koszul formulae Γ kij = 12 G kl (cid:18) ∂G jl ∂x i + ∂G li ∂x j − ∂G ij ∂x l (cid:19) for the Christoffel symbols of M × R , we can conclude that Γ kij = 0 whenever ( d + 1) ∈ { i, j, k } .Consider now a geodesic c ∗ in the product manifold M × R . Since its coordinate functions satisfythe second-order differential equations c ∗ k (cid:48)(cid:48) + (cid:0) Γ kij ◦ c ∗ (cid:1) c ∗ i (cid:48) c ∗ j (cid:48) = 0 and the Christoffel symbols have the previously mentioned properties, c ∗ ( d +1) (cid:48)(cid:48) = 0 follows. Thusthe last component of c ∗ is affine. Furthermore, by the behavior of the Christoffel symbols andby the geodesic differential equation again, the first projection of c ∗ results in a geodesic of M .Therefore any geodesic c ∗ connecting the points ( x , y ) and ( x , y ) of the product manifold M × R can be parametrized as c ∗ ( t ) = (cid:0) c ( t ) , (1 − t ) y + ty (cid:1) , SANDWICH WITH SEGMENT CONVEXITY 7 where the geodesic c of M connects the points c (0) = x and c (1) = x . Note also that thisparametrization is a global one in case of Cartan–Hadamard manifolds. This shows that the productmanifold M × R is the vertical extension of M .In particular, the vertical extension is a simply connected and complete manifold, as well. Nowwe show that its sectional curvature is nonpositive. Let σ ⊂ T ( x,y ) ( M × R ) = T x M ⊕ R be anarbitrary plane, and let b , b be a base in σ . If T x M ∩ σ = { } , then dim( T ( x,y ) ( M × R )) = d + 2 ,which is a contradiction. Thus dim( T x M ∩ σ ) ≥ , and we may assume that the second directcomponent of b is zero. Then b has a direct decomposition b = b + b . Recall that the sign ofthe sectional curvature depends only on the sign of R ( b , b , b , b ) , where R is the Riemanniancurvature tensor. Since its components can be obtained by R ijkl = (cid:18) ∂ Γ mjk ∂x i − ∂ Γ mik ∂x j + Γ rjk Γ mir − Γ rik Γ mjr (cid:19) G lm , we arrive at R ijkl = 0 provided that ( d + 1) ∈ { i, j, k, l } . Thus R vanishes if one of the argumentscontains b . Therefore, R ( b , b , b , b ) = R ( b + b , b , b , b + b ) = R ( b , b , b , b ) + R ( b , b , b , b )+ R ( b , b , b , b ) + R ( b , b , b , b ) = R ( b , b , b , b ) ≤ , since the second direct components of b and b are zero and the sectional curvature of M isnonpositive. This completes the proof. (cid:3) For more details on Cartan–Hadamard manifolds, we recommend the book of Jost [8]. As directconsequences of Theorem 1 and Theorem 2, we can formulate the next two corollaries.
Corollary 1.
Let D be a convex set in a Cartan–Hadamard manifold M whose vertical extension isdrop complete. If, for all n ∈ N , x , . . . , x n ∈ D , x ∈ conv { x , . . . , x n } , and for all t ∈ [0 , , thefunctions f, g : D → R satisfy (2) where c : [0 , → R is the geodesic segment joining x = c (0) and x = c (1) , then there exists a convex function ϕ : D → R fulfilling f ≤ ϕ ≤ g . Corollary 2.
Keeping the conditions of the previous theorem, assume that the Carathéodory num-ber of the vertical extension is κ . Then the size of the involved convex hull can be reduced to n ≤ κ . Cartan–Hadamard manifolds and Euclidean spaces are quite “close” relatives. Hence one mayexpect that the Carathéodory number of a Cartan–Hadamard manifold M , accordingly to the Eu-clidean case, can be expressed as κ = dim( M ) + 1 . However, as far as we know, this is still anopen problem posed by Ledyaev, Treiman, and Zhu [10].4. T HE EXOTIC BEHAVIOR OF CONVEX HULLS
The aim of this section is to prove, that drop completeness of the vertical extension cannot bechanged to drop completeness of the underlying system in Theorem 1 and Theorem 2. The mainreason is the exotic behavior of convex hulls: It may occur that the convex hull of three pointsin a Cartan–Hadamard manifold is not contained in a two dimensional submanifold. To constructsuch an example, let us recall here some basic facts in hyperbolic geometry. For references, seethe book of Ratcliffe [11].The hyperbolic plane, denoted by H in the forthcomings, is a two dimensional Cartan–Hadamardmanifold with constant sectional curvature − . We will use two of its several models. The first M. BESSENYEI, D. CS. KERTÉSZ, AND R. L. LOVAS one is the Beltrami–Klein model (known also as the Cayley–Klein model): Here the plane is theopen unit disc, and the lines are its Euclidean chord segments. The distance of this model will notbe used.The second model is the Poincaré half-plane model. The plane is the upper open Cartesianhalf-plane R × R + ; lines are either circles with center on the boundary line or vertical Euclideanhalf-lines. Its metric plays a key role in our investigation. The distance of a = ( a , a ) and b = ( b , b ) is given by(3) d ( a, b ) = 2 ln (cid:112) ( a − b ) + ( a − b ) + (cid:112) ( a − b ) + ( a + b ) √ a b . In particular, if a = b , this formula can be simplified (which will be quite convenient for us):(4) d ( a, b ) = | ln a − ln b | . If ( x , y ) and ( x , y ) are points of the vertical extension H × R , and c : [0 , → H is the uniquegeodesic fulfilling c (0) = x and c (1) = x , then the unique geodesic segment c ∗ : [0 , → H which connects ( x , y ) and ( x , y ) is given by c ∗ ( t ) = (cid:0) c ( t ) , (1 − t ) y + ty (cid:1) . Since geodesics are of constant speed, we have d ( x , x ) = td ( x , x ) for all t ∈ [0 , , where x = c ( t ) . Thus we can reconstruct the points ( x, y ) between ( x , y ) and ( x , y ) from x as(5) ( x, y ) = (cid:16) x, y + d ( x , x ) d ( x , x ) ( y − y ) (cid:17) . Theorem 3.
The hyperbolic plane is drop complete, while its vertical extension is not.Proof.
The Beltrami–Klein model shows, that the convex structure of H can be identified with theconvex structure of the open disc inherited from R . In particular, the drop representation is validin H .Now we prove, that the vertical extension H × R is not drop complete. We will illustrate itusing the convex hull of the points A = ((0 , , B = ((4 , , C = (( − , , . Consider their projections a, b, c and the additional points p, q, r on H : a = (0 , , b = (4 , , c = ( − , p = (1 , , q = ( − , , r = (0 , √ . It is immediate to check that p ∈ [ a, b ] and q ∈ [ a, c ] , furthermore r ∈ [ b, c ] hold (the segmentshere are meant in the hyperbolic geodesic sense: they are arcs of circles). Finally, we need theintersection of [ p, q ] and [ b, c ] , which turns out to be x = (0 , √ . The next figure shows thesechoices: SANDWICH WITH SEGMENT CONVEXITY 9
Let P and Q be the points on [ A, B ] and [ A, C ] in the vertical extension, whose first projectionsare p and q , respectively. Now we reconstruct their last coordinates from the projections. By thedistance formula (3), d ( a, b ) = 2 ln √
20 + √ √
15 = ln 3 and d ( a, x ) = 2 ln √ √ √
12 = ln 3 − ln 2 . Thus the second common projection of P and Q is obtained via (5) as ε := d ( a, x ) d ( a, b ) = 1 − ln 2ln 3 < . The estimation above can be checked even by hand. Moreover, the points of the segment [ P, Q ] share this common last component. Therefore, X := (cid:0) (0 , √ , ε (cid:1) ∈ [ X, Y ] ⊆ conv { A, B, C } . Clearly, R = ((0 , √ , ∈ [ B, C ] . Now we determine that point of the vertical extension,whose first projection is x , and belongs to the segment [ A, R ] . Using (4) and (5), its last componentturns out to be ε := ln √ − ln 3ln √ − ln 3 = ln 17 − − > . This estimation, with a bit more effort, can also be checked by hand. Thus, X := (cid:0) (0 , √ , ε (cid:1) ∈ [ A, R ] ⊆ (cid:91) { [ A, D ] | D ∈ [ B, C ] } . Since X (cid:54) = X , we can conclude that the drop representation involving { A } and [ B, C ] doesnot cover the entire convex hull of A, B, C , which was to be proved. (cid:3)
The Beltrami–Klein open sphere model and the first part of the argument show, that the geodesicconvex structure of hyperbolic space is compatible with the Euclidean convex sturcture of the openball.
In particular, the hyperbolic space is drop complete in any dimension, and its combinatorialinvariants coincide with the standard Euclidean ones. Using the Cartan–Hadamard theorem (orthe results of [2], it can be proved that these properties are also true for two dimensional Cartan–Hadamard manifolds.As we have already mentioned, the greatest advantage of Lemma 2 is that it can be implemented.In fact, the theorem above was conjectured via a computer algorithm. In what follows, we sketchbreafly its pseudo code.
We shall need two functions. The first one, geod calculates a point of a geodesic between twopoints: geod : ( H × R ) × ( H × R ) × R → H × R so that geod ( A, B,
0) = A and geod ( A, B,
1) = B hold. The function dist calculates the hyper-bolic distance of two points in H : dist ( a, b ) : H × H → R . The list
ConvexHull collects the points of the convex hull as an ordered list. Initially we put thepoints of the set whose convex hull is to be computed into
ConvexHull . ConvexHull [ i ] is the i th element in ConvexHull . Indexing starts with . The parameter iterations is the number ofiterations. Finally, the parameter res is the hyperbolic distance between points to be calculatedalong geodesics.The algorithm takes two points A and B from ConvexHull and adds the points of the geodesicfrom A to B with hyperbolic distance res from each other to ConvexHull . The point B is chosenso that repetitions are avoided. • The variables size and previoussize keep track of the number of points calculated inthe convex hull. Initially – size := | ConvexHull | . – previoussize := 0 . • The following loop is to be performed iterations many times. – For i = 0 , . . . , size , perform the following: ∗ For j = max( previoussize , i + 1) , . . . , size , perform the following: · A := ConvexHull [ i ] . · B := ConvexHull [ j ] . · d := dist ( A, B ) . · For k = 1 , , . . . while k · res < d ,add geod ( A, B, k · res /d ) to ConvexHull . – previoussize := size ; – size := | ConvexHull | .Using our algorithm, we can illustrate some points of conv { A, B, C } in the proof of Theorem 3.The first figure is the intersection of the approximation of the convex hull with the plane [ a, r ] × R :The second figure is the intersection of the approximation with [ p, q ] × R : SANDWICH WITH SEGMENT CONVEXITY 11
In each case, iterations = 2 and res = 0 . . In the concrete implementation, points areplotted whose distance from the planes is at most . .The program is available and freely downloadable from the homepage below: http://shrek.unideb.hu/˜ftzydk/convex/ Acknowledgement.
We wish to express our gratitude to professor S
ÁNDOR K RISTÁLY for thevaluable discussions on this topic. R
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