Fenchel Duality and a Separation Theorem on Hadamard Manifolds
FFenchel Duality and a Separation Theorem onHadamard Manifolds ∗ Ronny Bergmann † Roland Herzog † Maurício Silva Louzeiro † In this paper, we introduce a denition of Fenchel conjugate and Fenchel biconjugateon Hadamard manifolds based on the tangent bundle. Our denition overcomes theinconvenience that the conjugate depends on the choice of a certain point on the manifold,as previous denitions required. On the other hand, this new denition still possessesproperties known to hold in the Euclidean case. It even yields a broader interpretationof the Fenchel conjugate in the Euclidean case itself. Most prominently, our denitionof the Fenchel conjugate provides a Fenchel-Moreau Theorem for geodesically convex,proper, lower semicontinuous functions. In addition, this framework allows us to developa theory of separation of convex sets on Hadamard manifolds, and a strict separationtheorem is obtained.
Keywords. convex analysis, Fenchel conjugate function, Riemannian manifold, Hadamard manifold
AMS subject classications (MSC2010).
A central concept in convex analysis and related optimization algorithms is the notion of Fenchelduality. On the other hand, separation theorems for convex sets play an important role for thecharacterization of functions and their Fenchel conjugate. Among the vast references on these topics,we mention Bauschke, Combettes, 2011; Ekeland, Temam, 1999; Rockafellar, 1970; 1974; Zălinescu, 2002;Boţ, 2010; Brezis, 2011, all of which consider convex analysis and duality in vector spaces.The topic of optimization on Riemannian manifolds is currently receiving an increasing amount ofattention. We refer the reader to, e. g., Udrişte, 1994; Bačák, 2014; Absil, Mahony, Sepulchre, 2008; ∗ MSL is supported by a measure which is co-nanced by tax revenue based on the budget approved by the members of theSaxon state parliament. Financial support is gratefully acknowledged. † Technische Universität Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany ([email protected], , ORCID 0000-0001-8342-7218,[email protected], , OR-CID 0000-0003-2164-6575, [email protected], , ORCID 0000-0002-4755-3505). a r X i v : . [ m a t h . O C ] F e b . Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Boumal, 2020 and Rapcsák, 1997, Ch. 6 for background material. In this context, a theory of dualityon Riemannian manifolds has recently emerged, with particular emphasis on non-smooth problems(Bačák et al., 2016; Lellmann et al., 2013; Weinmann, Demaret, Storath, 2014) and related algorithms(Bergmann, Persch, Steidl, 2016; Bergmann, Chan, et al., 2016; Ferreira, Louzeiro, Prudente, 2020).To the best of our knowledge, there are up to now only two approaches to Fenchel duality on manifolds:one on Hadamard manifolds due to Ahmadi Kakavandi, Amini, 2010, and one on general Riemannianmanifolds proposed in Bergmann, Herzog, et al., 2021. In Ahmadi Kakavandi, Amini, 2010, the authorsintroduced a Fenchel conjugacy-like concept on complete CAT(0) spaces (usually called Hadamardspaces), using a quasilinearization in terms of distances as the duality product. For this purpose,a denition of 𝑜 -dual and 𝑜 -bidual was proposed, where 𝑜 is a point in the Hadamard space. Theauthors then show that this concept possesses several properties of the classical Fenchel conjugateon vector spaces, for instance the biconjugation theorem, and a generalization of the subdierentialcharacterization.Recently, we developed in Bergmann, Herzog, et al., 2021 a theory of duality on Riemannian mani-folds M by localizing the Fenchel conjugate similar to Bertsekas, 1978. As it was the case for AhmadiKakavandi, Amini, 2010, this concept also requires the choice of a base point 𝑚 on the manifold andit uses duality on the tangent space T 𝑚 M . Most of the analysis in Bergmann, Herzog, et al., 2021 isbased on properties of this tangent space as a vector space, and we can generalize many propertiesof the 𝑚 -Fenchel conjugate to Riemannian manifolds. We nally derived a generalization of theso-called Chambolle-Pock algorithm (Pock et al., 2009; Chambolle, Pock, 2011) for the minimizationof 𝑓 ( 𝑝 ) + 𝑔 ( Λ 𝑝 ) , where 𝑓 is dened on M , 𝑔 is dened on another Riemannian manifold N , and Λ : M → N . The algorithm generalizes a concept from Valkonen, 2014 and employs a linearization of Λ at a point 𝑚 as well as the 𝑛 -Fenchel conjugate for a second base point 𝑛 ∈ N . The convergenceresults rely on the convexity of the pull-back of 𝑔 onto the tangent space, i. e., on the convexity of thecomposition 𝑔 ◦ exp 𝑚 .In this paper we introduce a competing denition of a Fenchel conjugate on Hadamard manifolds. Ournew denition diers from the one in Bergmann, Herzog, et al., 2021 in two important ways. First, theconjugate of a function 𝐹 : M → R is dened on the entire cotangent bundle, not just on the cotangentspace at a particular base point. Second, we do not pull 𝐹 back to the tangent space. We also dene theFenchel biconjugate, which is—similar as in the competing approaches—again a function dened onthe manifold.Our new concept of duality possesses similar properties as those proved for the 𝑚 -Fenchel conjugate inBergmann, Herzog, et al., 2021. These include, in particular, the characterization of the subdierentialin terms of the Fenchel conjugate as well as the biconjucation theorem, also known as Fenchel–MoreauTheorem. The main dierence is that these results hold under more natural assumptions, notablygeodesic convexity of the function under consideration, rather than the convexity of its pull-back tothe tangent space.We would like to emphasize that our denition of Fenchel conjugate provides a broader understandingof the concept even for functions 𝑓 dened on a vector space 𝑉 . While clasically, the conjugate 𝑓 ∗ is afunction dened on 𝑉 ∗ , we obtain here a conjugate 𝑓 ∗ dened on the cotangent bundle 𝑉 × 𝑉 ∗ , forwhich 𝑓 ∗ ( , ·) agrees with the classical denition. cbnacbna
A central concept in convex analysis and related optimization algorithms is the notion of Fenchelduality. On the other hand, separation theorems for convex sets play an important role for thecharacterization of functions and their Fenchel conjugate. Among the vast references on these topics,we mention Bauschke, Combettes, 2011; Ekeland, Temam, 1999; Rockafellar, 1970; 1974; Zălinescu, 2002;Boţ, 2010; Brezis, 2011, all of which consider convex analysis and duality in vector spaces.The topic of optimization on Riemannian manifolds is currently receiving an increasing amount ofattention. We refer the reader to, e. g., Udrişte, 1994; Bačák, 2014; Absil, Mahony, Sepulchre, 2008; ∗ MSL is supported by a measure which is co-nanced by tax revenue based on the budget approved by the members of theSaxon state parliament. Financial support is gratefully acknowledged. † Technische Universität Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany ([email protected], , ORCID 0000-0001-8342-7218,[email protected], , OR-CID 0000-0003-2164-6575, [email protected], , ORCID 0000-0002-4755-3505). a r X i v : . [ m a t h . O C ] F e b . Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Boumal, 2020 and Rapcsák, 1997, Ch. 6 for background material. In this context, a theory of dualityon Riemannian manifolds has recently emerged, with particular emphasis on non-smooth problems(Bačák et al., 2016; Lellmann et al., 2013; Weinmann, Demaret, Storath, 2014) and related algorithms(Bergmann, Persch, Steidl, 2016; Bergmann, Chan, et al., 2016; Ferreira, Louzeiro, Prudente, 2020).To the best of our knowledge, there are up to now only two approaches to Fenchel duality on manifolds:one on Hadamard manifolds due to Ahmadi Kakavandi, Amini, 2010, and one on general Riemannianmanifolds proposed in Bergmann, Herzog, et al., 2021. In Ahmadi Kakavandi, Amini, 2010, the authorsintroduced a Fenchel conjugacy-like concept on complete CAT(0) spaces (usually called Hadamardspaces), using a quasilinearization in terms of distances as the duality product. For this purpose,a denition of 𝑜 -dual and 𝑜 -bidual was proposed, where 𝑜 is a point in the Hadamard space. Theauthors then show that this concept possesses several properties of the classical Fenchel conjugateon vector spaces, for instance the biconjugation theorem, and a generalization of the subdierentialcharacterization.Recently, we developed in Bergmann, Herzog, et al., 2021 a theory of duality on Riemannian mani-folds M by localizing the Fenchel conjugate similar to Bertsekas, 1978. As it was the case for AhmadiKakavandi, Amini, 2010, this concept also requires the choice of a base point 𝑚 on the manifold andit uses duality on the tangent space T 𝑚 M . Most of the analysis in Bergmann, Herzog, et al., 2021 isbased on properties of this tangent space as a vector space, and we can generalize many propertiesof the 𝑚 -Fenchel conjugate to Riemannian manifolds. We nally derived a generalization of theso-called Chambolle-Pock algorithm (Pock et al., 2009; Chambolle, Pock, 2011) for the minimizationof 𝑓 ( 𝑝 ) + 𝑔 ( Λ 𝑝 ) , where 𝑓 is dened on M , 𝑔 is dened on another Riemannian manifold N , and Λ : M → N . The algorithm generalizes a concept from Valkonen, 2014 and employs a linearization of Λ at a point 𝑚 as well as the 𝑛 -Fenchel conjugate for a second base point 𝑛 ∈ N . The convergenceresults rely on the convexity of the pull-back of 𝑔 onto the tangent space, i. e., on the convexity of thecomposition 𝑔 ◦ exp 𝑚 .In this paper we introduce a competing denition of a Fenchel conjugate on Hadamard manifolds. Ournew denition diers from the one in Bergmann, Herzog, et al., 2021 in two important ways. First, theconjugate of a function 𝐹 : M → R is dened on the entire cotangent bundle, not just on the cotangentspace at a particular base point. Second, we do not pull 𝐹 back to the tangent space. We also dene theFenchel biconjugate, which is—similar as in the competing approaches—again a function dened onthe manifold.Our new concept of duality possesses similar properties as those proved for the 𝑚 -Fenchel conjugate inBergmann, Herzog, et al., 2021. These include, in particular, the characterization of the subdierentialin terms of the Fenchel conjugate as well as the biconjucation theorem, also known as Fenchel–MoreauTheorem. The main dierence is that these results hold under more natural assumptions, notablygeodesic convexity of the function under consideration, rather than the convexity of its pull-back tothe tangent space.We would like to emphasize that our denition of Fenchel conjugate provides a broader understandingof the concept even for functions 𝑓 dened on a vector space 𝑉 . While clasically, the conjugate 𝑓 ∗ is afunction dened on 𝑉 ∗ , we obtain here a conjugate 𝑓 ∗ dened on the cotangent bundle 𝑉 × 𝑉 ∗ , forwhich 𝑓 ∗ ( , ·) agrees with the classical denition. cbnacbna page 2 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds An additional result in this paper is a theorem regarding the separation of convex sets on Hadamardmanifolds by geodesic hyperplanes in the cotangent bundle. This generalizes a well known separationtheorem from vector spaces to Hadamard spaces.The remainder of the paper is organized as follows. In Section 2 we recall a number of classical resultsfrom convex analysis in Hilbert spaces. In an eort to make the paper self-contained, we also brieystate the required concepts from dierential geometry and convex analysis on Hadamard manifolds.Section 3 is devoted to the development of the new notion of Fenchel conjugation for functions denedon Hadamard manifolds. Leveraging the concept, we extend some classical results from convex analysisto manifolds, like the Fenchel–Moreau Theorem (also known as the Biconjugation Theorem) and thecharacterization of the subdierential in terms of the conjugate function. In Section 4, we introduce atheory of separation of convex sets on Hadamard manifolds, which leads to a strict separation theorem.Finally, we give some conclusions and further remarks on future research in Section 5.
In this section we review some well known results from convex analysis in Hilbert spaces, which serveas the standard for comparison for the new results to be developed in Section 3. We emphasize thatthe results collected here are valid in more general contexts, but we do not strive for full generality.We also revisit necessary concepts from dierential geometry as well as the intersection of both topics,convex analysis on Riemannian manifolds, including its subdierential calculus.Throughout this paper we denote the extended line as R (cid:66) R ∪ {±∞} . We shall use the usualconvention −(−∞) = +∞ and −(+∞) = −∞ . In this subsection let X be a Hilbert space with inner product (· , ·) : X × X → R and duality pairing (cid:104)· , ·(cid:105) : X ∗ × X → R . Here, X ∗ denotes the dual space of X . For standard denitions like closedness,properness, lower semicontinuity (lsc) and convexity of a function 𝑓 : X → R , we refer the reader, e. g.,to the textbooks Rockafellar, 1970; Bauschke, Combettes, 2011. Denition 2.1.
The
Fenchel conjugate of a function 𝑓 : X → R is dened as the function 𝑓 ∗ : X ∗ → R such that 𝑓 ∗ ( 𝑥 ∗ ) (cid:66) sup 𝑥 ∈X (cid:8) (cid:104) 𝑥 ∗ , 𝑥 (cid:105) − 𝑓 ( 𝑥 ) (cid:9) . We recall some properties of the Fenchel conjugate function in Hilbert spaces in the following lemma.
Lemma 2.2 (Bauschke, Combettes, 2011, Ch. 13) . Let 𝑓 , 𝑔 : X → R be proper functions, 𝛼 ∈ R , 𝜆 > and 𝑧 ∈ X . Then the following statements hold. cbnacbna
Lemma 2.2 (Bauschke, Combettes, 2011, Ch. 13) . Let 𝑓 , 𝑔 : X → R be proper functions, 𝛼 ∈ R , 𝜆 > and 𝑧 ∈ X . Then the following statements hold. cbnacbna page 3 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds ( 𝑖 ) 𝑓 ∗ is convex and lsc. ( 𝑖𝑖 ) If 𝑓 ( 𝑥 ) ≤ 𝑔 ( 𝑥 ) for all 𝑥 ∈ X , then 𝑓 ∗ ( 𝑥 ∗ ) ≥ 𝑔 ∗ ( 𝑥 ∗ ) for all 𝑥 ∗ ∈ X ∗ . ( 𝑖𝑖𝑖 ) If 𝑔 ( 𝑥 ) = 𝑓 ( 𝑥 ) + 𝛼 for all 𝑥 ∈ X , then 𝑔 ∗ ( 𝑥 ∗ ) = 𝑓 ∗ ( 𝑥 ∗ ) − 𝛼 for all 𝑥 ∗ ∈ X ∗ . ( 𝑖𝑣 ) If 𝑔 ( 𝑥 ) = 𝜆𝑓 ( 𝑥 ) for all 𝑥 ∈ X , then 𝑔 ∗ ( 𝑥 ∗ ) = 𝜆𝑓 ∗ ( 𝑥 ∗ / 𝜆 ) for all 𝑥 ∗ ∈ X ∗ . ( 𝑣 ) If 𝑔 ( 𝑥 ) = 𝑓 ( 𝑥 + 𝑧 ) for all 𝑥 ∈ X , then 𝑔 ∗ ( 𝑥 ∗ ) = 𝑓 ∗ ( 𝑥 ∗ ) − (cid:104) 𝑥 ∗ , 𝑧 (cid:105) for all 𝑥 ∗ ∈ X ∗ . ( 𝑣𝑖 ) The
Fenchel–Young inequality holds, i. e., for all ( 𝑥, 𝑥 ∗ ) ∈ X × X ∗ we have (cid:104) 𝑥 ∗ , 𝑥 (cid:105) ≤ 𝑓 ( 𝑥 ) + 𝑓 ∗ ( 𝑥 ∗ ) . We now recall some results related to the denition of the subdierential of a proper function.
Denition 2.3 (Bauschke, Combettes, 2011, Def. 16.1) . Let 𝑓 : X → R be a proper function. Its subdif-ferential is dened as 𝜕𝑓 ( 𝑥 ) (cid:66) { 𝑥 ∗ ∈ X ∗ | 𝑓 ( 𝑧 ) ≥ 𝑓 ( 𝑥 ) + (cid:104) 𝑥 ∗ , 𝑧 − 𝑥 (cid:105) for all 𝑧 ∈ X} . Theorem 2.4 (Bauschke, Combettes, 2011, Prop. 16.9) . Let 𝑓 : X → R be a proper function and 𝑥 ∈ X .Then 𝑥 ∗ ∈ 𝜕𝑓 ( 𝑥 ) holds if and only if 𝑓 ( 𝑥 ) + 𝑓 ∗ ( 𝑥 ∗ ) = (cid:104) 𝑥 ∗ , 𝑥 (cid:105) . The Fenchel biconjugate 𝑓 ∗∗ : X → R of a function 𝑓 : X → R is given by 𝑓 ∗∗ ( 𝑥 ) = ( 𝑓 ∗ ) ∗ ( 𝑥 ) = sup 𝑥 ∗ ∈X ∗ (cid:8) (cid:104) 𝑥 ∗ , 𝑥 (cid:105) − 𝑓 ∗ ( 𝑥 ∗ ) (cid:9) . (2.1)It satises 𝑓 ∗∗ ( 𝑥 ) ≤ 𝑓 ( 𝑥 ) for all 𝑥 ∈ X ; see for instance Bauschke, Combettes, 2011, Prop. 13.14.We conclude this section with the following result known as the Fenchel–Moreau or BiconjugationTheorem. Theorem 2.5 (Bauschke, Combettes, 2011, Thm. 13.32) . Given a proper function 𝑓 : X → R , theequality 𝑓 ∗∗ ( 𝑥 ) = 𝑓 ( 𝑥 ) holds for all 𝑥 ∈ X if and only if 𝑓 is lsc and convex. In this case 𝑓 ∗ is proper aswell. This section is devoted to the collection of necessary concepts from dierential geometry. For detailsconcerning the subsequent denitions, the reader may wish to consult do Carmo, 1992; Lee, 2003; Jost,2017. cbnacbna
Denition 2.3 (Bauschke, Combettes, 2011, Def. 16.1) . Let 𝑓 : X → R be a proper function. Its subdif-ferential is dened as 𝜕𝑓 ( 𝑥 ) (cid:66) { 𝑥 ∗ ∈ X ∗ | 𝑓 ( 𝑧 ) ≥ 𝑓 ( 𝑥 ) + (cid:104) 𝑥 ∗ , 𝑧 − 𝑥 (cid:105) for all 𝑧 ∈ X} . Theorem 2.4 (Bauschke, Combettes, 2011, Prop. 16.9) . Let 𝑓 : X → R be a proper function and 𝑥 ∈ X .Then 𝑥 ∗ ∈ 𝜕𝑓 ( 𝑥 ) holds if and only if 𝑓 ( 𝑥 ) + 𝑓 ∗ ( 𝑥 ∗ ) = (cid:104) 𝑥 ∗ , 𝑥 (cid:105) . The Fenchel biconjugate 𝑓 ∗∗ : X → R of a function 𝑓 : X → R is given by 𝑓 ∗∗ ( 𝑥 ) = ( 𝑓 ∗ ) ∗ ( 𝑥 ) = sup 𝑥 ∗ ∈X ∗ (cid:8) (cid:104) 𝑥 ∗ , 𝑥 (cid:105) − 𝑓 ∗ ( 𝑥 ∗ ) (cid:9) . (2.1)It satises 𝑓 ∗∗ ( 𝑥 ) ≤ 𝑓 ( 𝑥 ) for all 𝑥 ∈ X ; see for instance Bauschke, Combettes, 2011, Prop. 13.14.We conclude this section with the following result known as the Fenchel–Moreau or BiconjugationTheorem. Theorem 2.5 (Bauschke, Combettes, 2011, Thm. 13.32) . Given a proper function 𝑓 : X → R , theequality 𝑓 ∗∗ ( 𝑥 ) = 𝑓 ( 𝑥 ) holds for all 𝑥 ∈ X if and only if 𝑓 is lsc and convex. In this case 𝑓 ∗ is proper aswell. This section is devoted to the collection of necessary concepts from dierential geometry. For detailsconcerning the subsequent denitions, the reader may wish to consult do Carmo, 1992; Lee, 2003; Jost,2017. cbnacbna page 4 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds
Suppose that M is an 𝑛 -dimensional connected, smooth manifold. The tangent space at 𝑝 ∈ M is avector space of dimension 𝑛 and it is denoted by T 𝑝 M . Its dual space is denoted by T ∗ 𝑝 M and it iscalled the cotangent space to M at 𝑝 . The duality product between 𝑋 ∈ T 𝑝 M and 𝜉 ∈ T ∗ 𝑝 M is denotedby (cid:104) 𝜉 , 𝑋 (cid:105) = 𝜉 ( 𝑋 ) ∈ R .The disjoint union of all tangent respectively cotangent spaces, i. e., T M (cid:66) (cid:216) 𝑝 ∈M { 𝑝 } × T 𝑝 M and T ∗ M (cid:66) (cid:216) 𝑝 ∈M { 𝑝 } × T ∗ 𝑝 M is called the tangent bundle respectively the cotangent bundle of M . Both are smooth manifolds ofdimension 2 𝑛 .We suppose that M is equipped with a Riemannian metric, i. e., a smoothly varying family of innerproducts on the tangent spaces T 𝑝 M . The metric at 𝑝 ∈ M is denoted by (· , ·) 𝑝 : T 𝑝 M × T 𝑝 M → R and we write (cid:107)·(cid:107) 𝑝 for the associated norm in T 𝑝 M . For simplicity we shall omit the index 𝑝 when noambiguity arises. The Riemannian metric furnishes a linear bijective correspondence between thetangent and cotangent spaces via the Riesz map and its inverse, the so-called musical isomorphisms ;see Lee, 2003, Ch. 8. They are dened as ♭ : T 𝑝 M (cid:51) 𝑋 ↦→ 𝑋 ♭ ∈ T ∗ 𝑝 M , (cid:104) 𝑋 ♭ , 𝑌 (cid:105) = ( 𝑋 , 𝑌 ) 𝑝 , for all 𝑌 ∈ T 𝑝 M , (2.2)and its inverse, ♯ : T ∗ 𝑝 M (cid:51) 𝜉 ↦→ 𝜉 ♯ ∈ T 𝑝 M , ( 𝜉 ♯ , 𝑌 ) 𝑝 = (cid:104) 𝜉 , 𝑌 (cid:105) , for all 𝑌 ∈ T 𝑝 M . (2.3)The ♯ -isomorphism further introduces an inner product and an associated norm on the cotangentspace T ∗ 𝑝 M , which we will also denote by (· , ·) 𝑝 and (cid:107)·(cid:107) 𝑝 , since it is clear which inner product ornorm we refer to based on the respective arguments.The tangent vector of a curve 𝛾 : 𝐼 → M dened on some open interval 𝐼 ⊆ R is denoted by (cid:164) 𝛾 ( 𝑡 ) .A curve is said to be geodesic if ∇ (cid:164) 𝛾 ( 𝑡 ) (cid:164) 𝛾 ( 𝑡 ) = 𝑡 ∈ 𝐼 , where ∇ denotes the Levi-Cevitaconnection, cf. do Carmo, 1992, Ch. 2 or Lee, 2018, Thm. 4.24. As a consequence, geodesic curves haveconstant speed. We say that a geodesic 𝛾 : [ , ] ⊂ R → M connects 𝑝 to 𝑞 if 𝛾 ( ) = 𝑝 and 𝛾 ( ) = 𝑞 holds. Notice that a geodesic connecting 𝑝 to 𝑞 need not always exist, and if it exists, it need not beunique. If a geodesic connecting 𝑝 to 𝑞 exists, there also exists a shortest geodesic among them, whichmay in turn not be unique. If it is, we denote the unique shortest geodesic connecting 𝑝 to 𝑞 by 𝛾 (cid:78) 𝑝,𝑞 .Moreover, given ( 𝑝, 𝑋 ) ∈ T M , we denote by 𝛾 𝑝,𝑋 : 𝐼 → M , with 𝐼 ⊆ R being a suitable open intervalcontaining 0, the geodesic starting at 𝑝 with (cid:164) 𝛾 𝑝,𝑋 ( ) = 𝑋 . We denote the subset of T 𝑝 M for whichthese geodesics are well dened until 𝑡 = G 𝑝 . Recall that a Riemannian manifold M is said to be(geodesically) complete if G 𝑝 = T 𝑝 M holds for some, and equivalently for all 𝑝 ∈ M .The Riemannian distance between 𝑝 and 𝑞 in M , dened as the inmum of the length over all piecewisesmooth curve segments from 𝑝 to 𝑞 , is denoted by 𝑑 ( 𝑝, 𝑞 ) . The metric topology it induces agrees withthe original topology on M . By the Hopf-Rinow theorem, M is geodesically complete if and only if itcomplete in the sense of metric spaces.The exponential map is dened as the function exp 𝑝 : G 𝑝 → M with exp 𝑝 𝑋 (cid:66) 𝛾 𝑝,𝑋 ( ) . Notethat exp 𝑝 ( 𝑡𝑋 ) = 𝛾 𝑝,𝑋 ( 𝑡 ) holds for every 𝑡 ∈ [ , ] . We further introduce the set G (cid:48) 𝑝 ⊆ T 𝑝 M as cbnacbna
Suppose that M is an 𝑛 -dimensional connected, smooth manifold. The tangent space at 𝑝 ∈ M is avector space of dimension 𝑛 and it is denoted by T 𝑝 M . Its dual space is denoted by T ∗ 𝑝 M and it iscalled the cotangent space to M at 𝑝 . The duality product between 𝑋 ∈ T 𝑝 M and 𝜉 ∈ T ∗ 𝑝 M is denotedby (cid:104) 𝜉 , 𝑋 (cid:105) = 𝜉 ( 𝑋 ) ∈ R .The disjoint union of all tangent respectively cotangent spaces, i. e., T M (cid:66) (cid:216) 𝑝 ∈M { 𝑝 } × T 𝑝 M and T ∗ M (cid:66) (cid:216) 𝑝 ∈M { 𝑝 } × T ∗ 𝑝 M is called the tangent bundle respectively the cotangent bundle of M . Both are smooth manifolds ofdimension 2 𝑛 .We suppose that M is equipped with a Riemannian metric, i. e., a smoothly varying family of innerproducts on the tangent spaces T 𝑝 M . The metric at 𝑝 ∈ M is denoted by (· , ·) 𝑝 : T 𝑝 M × T 𝑝 M → R and we write (cid:107)·(cid:107) 𝑝 for the associated norm in T 𝑝 M . For simplicity we shall omit the index 𝑝 when noambiguity arises. The Riemannian metric furnishes a linear bijective correspondence between thetangent and cotangent spaces via the Riesz map and its inverse, the so-called musical isomorphisms ;see Lee, 2003, Ch. 8. They are dened as ♭ : T 𝑝 M (cid:51) 𝑋 ↦→ 𝑋 ♭ ∈ T ∗ 𝑝 M , (cid:104) 𝑋 ♭ , 𝑌 (cid:105) = ( 𝑋 , 𝑌 ) 𝑝 , for all 𝑌 ∈ T 𝑝 M , (2.2)and its inverse, ♯ : T ∗ 𝑝 M (cid:51) 𝜉 ↦→ 𝜉 ♯ ∈ T 𝑝 M , ( 𝜉 ♯ , 𝑌 ) 𝑝 = (cid:104) 𝜉 , 𝑌 (cid:105) , for all 𝑌 ∈ T 𝑝 M . (2.3)The ♯ -isomorphism further introduces an inner product and an associated norm on the cotangentspace T ∗ 𝑝 M , which we will also denote by (· , ·) 𝑝 and (cid:107)·(cid:107) 𝑝 , since it is clear which inner product ornorm we refer to based on the respective arguments.The tangent vector of a curve 𝛾 : 𝐼 → M dened on some open interval 𝐼 ⊆ R is denoted by (cid:164) 𝛾 ( 𝑡 ) .A curve is said to be geodesic if ∇ (cid:164) 𝛾 ( 𝑡 ) (cid:164) 𝛾 ( 𝑡 ) = 𝑡 ∈ 𝐼 , where ∇ denotes the Levi-Cevitaconnection, cf. do Carmo, 1992, Ch. 2 or Lee, 2018, Thm. 4.24. As a consequence, geodesic curves haveconstant speed. We say that a geodesic 𝛾 : [ , ] ⊂ R → M connects 𝑝 to 𝑞 if 𝛾 ( ) = 𝑝 and 𝛾 ( ) = 𝑞 holds. Notice that a geodesic connecting 𝑝 to 𝑞 need not always exist, and if it exists, it need not beunique. If a geodesic connecting 𝑝 to 𝑞 exists, there also exists a shortest geodesic among them, whichmay in turn not be unique. If it is, we denote the unique shortest geodesic connecting 𝑝 to 𝑞 by 𝛾 (cid:78) 𝑝,𝑞 .Moreover, given ( 𝑝, 𝑋 ) ∈ T M , we denote by 𝛾 𝑝,𝑋 : 𝐼 → M , with 𝐼 ⊆ R being a suitable open intervalcontaining 0, the geodesic starting at 𝑝 with (cid:164) 𝛾 𝑝,𝑋 ( ) = 𝑋 . We denote the subset of T 𝑝 M for whichthese geodesics are well dened until 𝑡 = G 𝑝 . Recall that a Riemannian manifold M is said to be(geodesically) complete if G 𝑝 = T 𝑝 M holds for some, and equivalently for all 𝑝 ∈ M .The Riemannian distance between 𝑝 and 𝑞 in M , dened as the inmum of the length over all piecewisesmooth curve segments from 𝑝 to 𝑞 , is denoted by 𝑑 ( 𝑝, 𝑞 ) . The metric topology it induces agrees withthe original topology on M . By the Hopf-Rinow theorem, M is geodesically complete if and only if itcomplete in the sense of metric spaces.The exponential map is dened as the function exp 𝑝 : G 𝑝 → M with exp 𝑝 𝑋 (cid:66) 𝛾 𝑝,𝑋 ( ) . Notethat exp 𝑝 ( 𝑡𝑋 ) = 𝛾 𝑝,𝑋 ( 𝑡 ) holds for every 𝑡 ∈ [ , ] . We further introduce the set G (cid:48) 𝑝 ⊆ T 𝑝 M as cbnacbna page 5 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds some open set such that exp 𝑝 : G (cid:48) 𝑝 → exp 𝑝 (G (cid:48) 𝑝 ) ⊆ M is a dieomorphism. The logarithmic map isdened as the inverse of the exponential map, i. e., log 𝑝 : exp 𝑝 (G (cid:48) 𝑝 ) → G (cid:48) 𝑝 ⊆ T 𝑝 M .In the particular case where the sectional curvature of the manifold is nonpositive everywhere, thegeodesics connecting any two distinct points exist and are unique. If furthermore, the manifold issimply connected and complete, it is called a Hadamard manifold ; see Bačák, 2014, p.10. In this case,the exponential and logarithmic maps are dened globally, i. e., G 𝑝 = T 𝑝 M holds for all 𝑝 ∈ M . Theseproperties of Hadamard manifolds make these spaces particularly amenable for the study of convexityproperties. Throughout this subsection, M is assumed to be a Hadamard manifold and we recall the basic conceptsof convex analysis on M . The central idea is to replace straight lines in the denition of convex sets inHilbert spaces by geodesics. Denition 2.6 (Sakai, 1996, Def. IV.5.9, Def. IV.5.1) . ( 𝑖 ) A function 𝐹 : M → R is proper if dom 𝐹 (cid:66) { 𝑝 ∈ M | 𝐹 ( 𝑝 ) < ∞} ≠ ∅ and 𝐹 ( 𝑝 ) > −∞ holds forall 𝑝 ∈ M . ( 𝑖𝑖 ) A function 𝐹 : M → R is convex if, for all 𝑝, 𝑞 ∈ M , the composition 𝐹 ◦ 𝛾 (cid:78) 𝑝,𝑞 : [ , ] ⊂ R → R isa convex function on [ , ] in the classical sense. ( 𝑖𝑖𝑖 ) The epigraph of a function 𝐹 : M → R is dened as epi 𝐹 (cid:66) {( 𝑝, 𝛼 ) ∈ M × R | 𝐹 ( 𝑝 ) ≤ 𝛼 } . (2.4) ( 𝑖𝑣 ) A proper function 𝐹 : M → R is called lower semicontinuous (lsc) if epi 𝐹 is closed. ( 𝑣 ) A subset
C ⊆ M is said to be convex if for any two points 𝑝, 𝑞 ∈ C , the unique geodesic of M connecting 𝑝 to 𝑞 lies completely in C . We now recall the notion of the subdierential of a geodesically convex function.
Denition 2.7 (Ferreira, Oliveira, 1998, Udrişte, 1994, Def. 3.4.4) . The subdierential 𝜕𝐹 at a point 𝑝 ∈M of a proper, convex function 𝐹 : M → R is given by 𝜕𝐹 ( 𝑝 ) (cid:66) (cid:8) 𝜉 ∈ T ∗ 𝑝 M (cid:12)(cid:12) 𝐹 ( 𝑞 ) ≥ 𝐹 ( 𝑝 ) + (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) for all 𝑞 ∈ M (cid:9) . When
C ⊆ M is nonempty, convex and closed, it was proved in Ferreira, Oliveira, 2002 that for eachpoint 𝑝 ∈ M , there is a unique point ˆ 𝑝 ∈ C satisfying 𝑑 ( 𝑝, ˆ 𝑝 ) ≤ 𝑑 ( 𝑝, 𝑞 ) for all 𝑞 ∈ C . In this case, ˆ 𝑝 iscalled the projection of 𝑝 onto C and it is denoted by proj C ( 𝑝 ) . We require the following result fromFerreira, Oliveira, 2002, Cor. 3.1. cbnacbna
C ⊆ M is nonempty, convex and closed, it was proved in Ferreira, Oliveira, 2002 that for eachpoint 𝑝 ∈ M , there is a unique point ˆ 𝑝 ∈ C satisfying 𝑑 ( 𝑝, ˆ 𝑝 ) ≤ 𝑑 ( 𝑝, 𝑞 ) for all 𝑞 ∈ C . In this case, ˆ 𝑝 iscalled the projection of 𝑝 onto C and it is denoted by proj C ( 𝑝 ) . We require the following result fromFerreira, Oliveira, 2002, Cor. 3.1. cbnacbna page 6 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Theorem 2.8.
Suppose that
C ⊆ M a nonempty, convex and closed set and 𝑝 ∈ M . Then the followinginequality holds, (cid:0) log proj C ( 𝑝 ) 𝑝 , log proj C ( 𝑝 ) 𝑞 (cid:1) ≤ for all 𝑞 ∈ C . Corollary 2.9.
Let 𝐹 : M → R be a proper lsc convex function and ( 𝑝, 𝑠 ) ∉ epi 𝐹 . Then the projection proj epi 𝐹 ( 𝑝, 𝑠 ) (cid:67) ( ˆ 𝑝, ˆ 𝑠 ) exists and the following inequality holds, (cid:0) log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑞 (cid:1) + ( 𝑠 − ˆ 𝑠 ) ( 𝑟 − ˆ 𝑠 ) ≤ for all ( 𝑞, 𝑟 ) ∈ epi 𝐹 .
Proof.
Since 𝐹 : M → R is a proper lsc convex function, epi 𝐹 ⊂ M × R is a nonempty closed convexset, where M × R is equipped with the product metric. Hence, using Theorem 2.8 with C = epi 𝐹 weget the desired inequality. (cid:3) A geodesic triangle Δ ( 𝑝 , 𝑝 , 𝑝 ) of a Hadamard manifold is the set consisting of three distinct points 𝑝 , 𝑝 , 𝑝 called the vertices and three geodesics 𝛾 (cid:80) 𝑝 ,𝑝 , 𝛾 (cid:80) 𝑝 ,𝑝 , 𝛾 (cid:80) 𝑝 ,𝑝 . The proof of the following theoremcan be found in Ferreira, Oliveira, 2002, Thm. 2.2. Theorem 2.10.
Suppose that Δ ( 𝑝 , 𝑝 , 𝑝 ) a geodesic triangle. Then, 𝑑 ( 𝑝 𝑖 (cid:9) , 𝑝 𝑖 ) − (cid:0) log 𝑝 𝑖 𝑝 𝑖 (cid:9) , log 𝑝 𝑖 𝑝 𝑖 ⊕ (cid:1) + 𝑑 ( 𝑝 𝑖 ⊕ , 𝑝 𝑖 ) ≤ 𝑑 ( 𝑝 𝑖 (cid:9) , 𝑝 𝑖 ⊕ ) , (2.5) (cid:0) log 𝑝 𝑖 𝑝 𝑖 (cid:9) , log 𝑝 𝑖 𝑝 𝑖 ⊕ (cid:1) + (cid:0) log 𝑝 𝑖 ⊕ 𝑝 𝑖 (cid:9) , log 𝑝 𝑖 ⊕ 𝑝 𝑖 (cid:1) ≥ 𝑑 ( 𝑝 𝑖 ⊕ , 𝑝 𝑖 ) , (2.6) for 𝑖 = , , , where the indices 𝑖 (cid:9) and 𝑖 ⊕ are meant modulo . In this section we introduce new denitions of the Fenchel conjugate and Fenchel biconjugate forextended real-valued functions dened on Hadamard manifolds. Using these denitions, we can extendfundamental properties from the Euclidean to the Riemannian setting.
We begin with a new denition of the Fenchel conjugate function on Hadamard manifolds.
Denition 3.1.
Let 𝐹 : M → R . The Fenchel conjugate of 𝐹 is the function 𝐹 ∗ : T ∗ M → R dened by 𝐹 ∗ ( 𝑝, 𝜉 ) (cid:66) sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) for ( 𝑝, 𝜉 ) ∈ T ∗ M . (3.1) cbnacbna
Let 𝐹 : M → R . The Fenchel conjugate of 𝐹 is the function 𝐹 ∗ : T ∗ M → R dened by 𝐹 ∗ ( 𝑝, 𝜉 ) (cid:66) sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) for ( 𝑝, 𝜉 ) ∈ T ∗ M . (3.1) cbnacbna page 7 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds As was mentioned in the introduction, this denition diers from Bergmann, Herzog, et al., 2021,Def. 3.1 in two important ways. First, 𝐹 ∗ is dened on the entire cotangent bundle, not just on thecotangent space at a particular base point. Second, we do not pull 𝐹 back to the tangent space. Remark 3.2.
Suppose that 𝐹 : M → R is a proper function. Since (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) = −∞ holds for all 𝑞 ∉ dom 𝐹 , we have 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈ dom 𝐹 (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) for all ( 𝑝, 𝜉 ) ∈ T ∗ M . Remark 3.3.
Note that for each ( 𝑝, 𝜉 ) ∈ T ∗ M , 𝐹 ∗ ( 𝑝, 𝜉 ) agrees with the 𝑝 -Fenchel conjugate 𝐹 ∗ 𝑝 ( 𝜉 ) introduced in Bergmann, Herzog, et al., 2021, Def. 3.1. For convenience, let us recall that 𝐹 ∗ 𝑝 : T ∗ 𝑝 M → R isdened as 𝐹 ∗ 𝑝 ( 𝜉 ) = sup 𝑋 ∈T 𝑝 M (cid:8) (cid:104) 𝜉 , 𝑋 (cid:105) − 𝐹 ( exp 𝑝 𝑋 ) (cid:9) for 𝜉 ∈ T ∗ 𝑝 M . The equality 𝐹 ∗ ( 𝑝, 𝜉 ) = 𝐹 ∗ 𝑝 ( 𝜉 ) follows immediately from the relation 𝑋 = log 𝑝 𝑞 ⇔ 𝑞 = exp 𝑝 𝑋 onHadamard manifolds.We also observe that in case M is the Euclidean space R 𝑛 , Denition 3.1 becomes 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈ R 𝑛 (cid:8) (cid:104) 𝜉 , 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) − (cid:104) 𝜉 , 𝑝 (cid:105) = 𝐹 ∗ ( 𝜉 ) − (cid:104) 𝜉 , 𝑝 (cid:105) = 𝐹 ∗ ( 𝑝 + 𝜉 ) for all ( 𝑝, 𝜉 ) ∈ R 𝑛 × R 𝑛 , (3.2) where the last equality is due to Lemma 2.2 ( 𝑣 ) . Hence, at 𝑝 = we recover the classical (Euclidean)conjugate from Denition 2.1 with X = R 𝑛 . Example 3.4.
Let 𝑝 ∈ M be arbitrary but xed and let 𝐹 : M → R be dened by 𝐹 ( 𝑞 ) = 𝑑 ( 𝑝, 𝑞 ) . Dueto the fact that 𝑑 ( 𝑝, 𝑞 ) = (cid:107) log 𝑝 𝑞 (cid:107) holds, we obtain from Denition 3.1 the following representation of 𝐹 ∗ : 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) for ( 𝑝, 𝜉 ) ∈ T ∗ M . (3.3) For every 𝜉 ∈ T ∗ 𝑝 M with (cid:107) 𝜉 (cid:107) ≤ , the following inequalities hold: = (cid:104) 𝜉 , log 𝑝 𝑝 (cid:105) − (cid:107) log 𝑝 𝑝 (cid:107) ≤ sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) ≤ sup 𝑞 ∈M (cid:8) (cid:107) 𝜉 (cid:107) (cid:107) log 𝑝 𝑞 (cid:107) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) ≤ . Hence, (3.3) implies 𝐹 ∗ ( 𝑝, 𝜉 ) = whenever (cid:107) 𝜉 (cid:107) ≤ . On the other hand, if (cid:107) 𝜉 (cid:107) > holds, then 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) = sup 𝑋 ∈T 𝑝 M (cid:8) (cid:104) 𝜉 , 𝑋 (cid:105) − (cid:107) 𝑋 (cid:107) (cid:9) ≥ sup 𝜆 > (cid:8) (cid:104) 𝜉 , 𝜆 𝜉 ♯ (cid:105) − (cid:107) 𝜆 𝜉 ♯ (cid:107) (cid:9) = sup 𝜆 > (cid:8) 𝜆 ((cid:107) 𝜉 (cid:107) − (cid:107) 𝜉 (cid:107)) (cid:9) = +∞ . cbnacbna
Let 𝑝 ∈ M be arbitrary but xed and let 𝐹 : M → R be dened by 𝐹 ( 𝑞 ) = 𝑑 ( 𝑝, 𝑞 ) . Dueto the fact that 𝑑 ( 𝑝, 𝑞 ) = (cid:107) log 𝑝 𝑞 (cid:107) holds, we obtain from Denition 3.1 the following representation of 𝐹 ∗ : 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) for ( 𝑝, 𝜉 ) ∈ T ∗ M . (3.3) For every 𝜉 ∈ T ∗ 𝑝 M with (cid:107) 𝜉 (cid:107) ≤ , the following inequalities hold: = (cid:104) 𝜉 , log 𝑝 𝑝 (cid:105) − (cid:107) log 𝑝 𝑝 (cid:107) ≤ sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) ≤ sup 𝑞 ∈M (cid:8) (cid:107) 𝜉 (cid:107) (cid:107) log 𝑝 𝑞 (cid:107) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) ≤ . Hence, (3.3) implies 𝐹 ∗ ( 𝑝, 𝜉 ) = whenever (cid:107) 𝜉 (cid:107) ≤ . On the other hand, if (cid:107) 𝜉 (cid:107) > holds, then 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:9) = sup 𝑋 ∈T 𝑝 M (cid:8) (cid:104) 𝜉 , 𝑋 (cid:105) − (cid:107) 𝑋 (cid:107) (cid:9) ≥ sup 𝜆 > (cid:8) (cid:104) 𝜉 , 𝜆 𝜉 ♯ (cid:105) − (cid:107) 𝜆 𝜉 ♯ (cid:107) (cid:9) = sup 𝜆 > (cid:8) 𝜆 ((cid:107) 𝜉 (cid:107) − (cid:107) 𝜉 (cid:107)) (cid:9) = +∞ . cbnacbna page 8 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Therefore, the Fenchel conjugate 𝐹 ∗ : T ∗ M → R of 𝐹 = 𝑑 ( 𝑝, ·) is given by 𝐹 ∗ ( 𝑝, 𝜉 ) = (cid:40) if (cid:107) 𝜉 (cid:107) ≤ , +∞ if (cid:107) 𝜉 (cid:107) > . Example 3.5.
Let 𝑝 ∈ M be arbitrary but xed and let 𝐹 : M → R be dened by 𝐹 ( 𝑞 ) = 𝑑 ( 𝑝, 𝑞 ) .Then we have 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:110) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − (cid:107) log 𝑝 𝑞 (cid:107) (cid:111) = sup 𝑋 ∈T 𝑝 M (cid:110) (cid:104) 𝜉 , 𝑋 (cid:105) − (cid:107) 𝑋 (cid:107) (cid:111) = (cid:107) 𝜉 (cid:107) . In particular, we obtain 𝐹 ∗ ( 𝑝, 𝜉 ) = 𝐹 ∗ ( 𝑝, − 𝜉 ) for all ( 𝑝, 𝜉 ) ∈ T ∗ M . In addition, the following propertyholds: 𝐹 ∗ (cid:0) 𝑝, [ log 𝑝 𝑞 ] ♭ (cid:1) = (cid:107) [ log 𝑝 𝑞 ] ♭ (cid:107) = 𝑑 ( 𝑝, 𝑞 ) = 𝐹 ( 𝑞 ) for all 𝑞 ∈ M . In comparison with the classical conjugate on R 𝑛 , it appears unusual that 𝐹 ∗ from Denition 3.1depends on two arguments, 𝑝 and 𝜉 . One might expect there to be some redundancy in the denition.Indeed, this redundancy has already been observed in (3.2) for the Euclidean setting. We now exploreit in the Riemannian case.To this end, we consider the following equivalence relation on the cotangent bundle T ∗ M : ( 𝑝, 𝜉 ) ∼ ( 𝑝 (cid:48) , 𝜉 (cid:48) ) if and only if (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) = (cid:104) 𝜉 (cid:48) , log 𝑝 (cid:48) 𝑞 (cid:105) holds for all 𝑞 ∈ M . (3.4)The equivalence class of ( 𝑝, 𝜉 ) ∈ T ∗ M , denoted by [( 𝑝, 𝜉 )] , is [( 𝑝, 𝜉 )] = {( 𝑝 (cid:48) , 𝜉 (cid:48) ) ∈ T ∗ M | (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) = (cid:104) 𝜉 (cid:48) , log 𝑝 (cid:48) 𝑞 (cid:105) for all 𝑞 ∈ M} . (3.5)Note that 𝐹 ∗ ( 𝑝 (cid:48) , 𝜉 (cid:48) ) = 𝐹 ∗ ( 𝑝, 𝜉 ) holds for all ( 𝑝 (cid:48) , 𝜉 (cid:48) ) ∈ [( 𝑝, 𝜉 )] . Observe as well that when M is theEuclidean space R 𝑛 , the equivalence class of ( 𝑝, 𝜉 ) ∈ T ∗ M becomes [( 𝑝, 𝜉 )] = {( 𝑝 (cid:48) , 𝜉 (cid:48) ) ∈ R 𝑛 × R 𝑛 | (cid:104) 𝜉 , 𝑞 − 𝑝 (cid:105) = (cid:104) 𝜉 (cid:48) , 𝑞 − 𝑝 (cid:48) (cid:105) for all 𝑞 ∈ R 𝑛 } = {( 𝑝 (cid:48) , 𝜉 (cid:48) ) ∈ R 𝑛 × R 𝑛 | (cid:104) 𝜉 − 𝜉 (cid:48) , 𝑞 (cid:105) = (cid:104) 𝜉 , 𝑝 (cid:105) − (cid:104) 𝜉 (cid:48) , 𝑝 (cid:48) (cid:105) for all 𝑞 ∈ R 𝑛 } = {( 𝑝 (cid:48) , 𝜉 (cid:48) ) ∈ R 𝑛 × R 𝑛 | 𝜉 (cid:48) = 𝜉, (cid:104) 𝜉 (cid:48) , 𝑝 (cid:48) (cid:105) = (cid:104) 𝜉 , 𝑝 (cid:105)} = {( 𝑝 (cid:48) , 𝜉 ) ∈ R 𝑛 × R 𝑛 | (cid:104) 𝜉 , 𝑝 (cid:48) (cid:105) = (cid:104) 𝜉 , 𝑝 (cid:105)} , which describes a hyperplane in R 𝑛 when 𝜉 ≠
0. The following example illustrates that the equivalenceclass dened in (3.5) is, in general, not a singleton even in non-Euclidean manifolds.
Example 3.6.
We denote by M = P ( 𝑛 ) the cone of real, symmetric positive denite matrices of size 𝑛 × 𝑛 . Its tangent space (at any point) can be identied with S ( 𝑛 ) , the space of symmetric 𝑛 × 𝑛 -matrices.The manifold M is endowed with the ane invariant Riemannian metric, which at 𝐴 ∈ P ( 𝑛 ) is given by ( 𝑋 , 𝑌 ) 𝐴 (cid:66) trace ( 𝑋 𝐴 − 𝑌 𝐴 − ) for 𝑋 , 𝑌 ∈ T 𝐴 M . (3.6) cbnacbna
We denote by M = P ( 𝑛 ) the cone of real, symmetric positive denite matrices of size 𝑛 × 𝑛 . Its tangent space (at any point) can be identied with S ( 𝑛 ) , the space of symmetric 𝑛 × 𝑛 -matrices.The manifold M is endowed with the ane invariant Riemannian metric, which at 𝐴 ∈ P ( 𝑛 ) is given by ( 𝑋 , 𝑌 ) 𝐴 (cid:66) trace ( 𝑋 𝐴 − 𝑌 𝐴 − ) for 𝑋 , 𝑌 ∈ T 𝐴 M . (3.6) cbnacbna page 9 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds M is a Hadamard manifold; see for instance Lang, 1999, Ch. XII, Thm. 1.2, p. 325. When we identify thecotangent space with S ( 𝑛 ) via the duality (cid:104) 𝜉 , 𝑌 (cid:105) (cid:66) trace ( 𝜉 𝑌 ) , then the ’at’ isomorphism ♭ at 𝐴 is givenby 𝑋 ♭ = 𝐴 − 𝑋 𝐴 − (3.7) since (cid:104) 𝑋 ♭ , 𝑌 (cid:105) = trace ( 𝑋 ♭ 𝑌 ) = trace ( 𝐴 − 𝑋 𝐴 − 𝑌 ) = trace ( 𝑋 𝐴 − 𝑌 𝐴 − ) = ( 𝑋 , 𝑌 ) 𝐴 holds for all 𝑌 ∈ S ( 𝑛 ) .The logarithmic map log 𝐴 : M → T 𝐴 M is given by log 𝐴 𝐵 = 𝐴 / ln (cid:0) 𝐴 − / 𝐵 𝐴 − / (cid:1) 𝐴 / for 𝐴, 𝐵 ∈ M , (3.8) where · / and ln denote the matrix square root and matrix logarithm of symmetric positive denitematrices, respectively. We refer the reader, for instance, to Higham, 2008, Thms. 1.29 and 1.31.Suppose that 𝐴 ∈ P ( 𝑛 ) is arbitrary but xed and consider the particular cotangent vector 𝐴 ♭ = 𝐴 − ∈ S ( 𝑛 ) .Using (3.8) , we evaluate (cid:10) 𝐴 ♭ , log 𝐴 𝐵 (cid:11) = trace (cid:0) 𝐴 − 𝐴 / ln (cid:0) 𝐴 − / 𝐵 𝐴 − / (cid:1) 𝐴 / (cid:1) = trace ln (cid:0) 𝐴 − / 𝐵 𝐴 − / (cid:1) = trace ln 𝐴 − / + trace ln 𝐵 + trace ln 𝐴 − / = trace ln 𝐵 − trace ln 𝐴 for any 𝐵 ∈ P ( 𝑛 ) . Here we also used that trace ln ( 𝐶𝐷 ) = trace ln 𝐶 + trace ln 𝐷 holds for positive denitematrices 𝐶 and 𝐷 as well as ln 𝐶 − = − ln 𝐶 .Now choose any orthogonal matrix 𝑉 and set (cid:98) 𝐴 (cid:66) 𝑉 𝐴 𝑉 − . Then the same reasoning as above shows (cid:10) (cid:98) 𝐴 ♭ , log (cid:98) 𝐴 𝐵 (cid:11) = trace ln 𝐵 − trace ln (cid:98) 𝐴 = trace ln 𝐵 − trace ln 𝐴. The second equality follows from the fact that 𝐴 and (cid:98) 𝐴 have the same eigenvalues and thus the same istrue for ln 𝐴 and ln (cid:98) 𝐴 .We conclude that ( 𝐴, 𝐴 ♭ ) and ( (cid:98) 𝐴, (cid:98) 𝐴 ♭ ) belong to the same equivalence class w.r.t. the relation (3.4) . Therefore,the equivalence classes (3.5) are not, in general, singletons. The following results establishes a property which relates the Fenchel conjugate evaluated in elementsof T ∗ M whose base points are not necessarily the same. Proposition 3.7.
Let 𝐹 : M → R and 𝑝, 𝑝 (cid:48) ∈ M . Then the following inequality holds: 𝐹 ∗ (cid:0) 𝑝, [ log 𝑝 𝑝 (cid:48) ] ♭ (cid:1) ≥ 𝐹 ∗ (cid:0) 𝑝 (cid:48) , [− log 𝑝 (cid:48) 𝑝 ] ♭ (cid:1) + 𝑑 ( 𝑝, 𝑝 (cid:48) ) . Proof.
Consider the geodesic triangle Δ ( 𝑝, 𝑝 (cid:48) , 𝑞 ) with some 𝑞 ∈ M . Using (2.2) and (2.6) with 𝑝 𝑖 = 𝑝 (cid:48) , 𝑝 𝑖 ⊕ = 𝑝 and 𝑝 𝑖 (cid:9) = 𝑞 we can say that (cid:104)[ log 𝑝 𝑝 (cid:48) ] ♭ , log 𝑝 𝑞 (cid:105) + (cid:104)[ log 𝑝 (cid:48) 𝑝 ] ♭ , log 𝑝 (cid:48) 𝑞 (cid:105) = ( log 𝑝 𝑝 (cid:48) , log 𝑝 𝑞 ) + ( log 𝑝 (cid:48) 𝑝 , log 𝑝 (cid:48) 𝑞 ) ≥ 𝑑 ( 𝑝, 𝑝 (cid:48) ) . cbnacbna
Consider the geodesic triangle Δ ( 𝑝, 𝑝 (cid:48) , 𝑞 ) with some 𝑞 ∈ M . Using (2.2) and (2.6) with 𝑝 𝑖 = 𝑝 (cid:48) , 𝑝 𝑖 ⊕ = 𝑝 and 𝑝 𝑖 (cid:9) = 𝑞 we can say that (cid:104)[ log 𝑝 𝑝 (cid:48) ] ♭ , log 𝑝 𝑞 (cid:105) + (cid:104)[ log 𝑝 (cid:48) 𝑝 ] ♭ , log 𝑝 (cid:48) 𝑞 (cid:105) = ( log 𝑝 𝑝 (cid:48) , log 𝑝 𝑞 ) + ( log 𝑝 (cid:48) 𝑝 , log 𝑝 (cid:48) 𝑞 ) ≥ 𝑑 ( 𝑝, 𝑝 (cid:48) ) . cbnacbna page 10 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Hence, we obtain (cid:104)[ log 𝑝 𝑝 (cid:48) ] ♭ , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) ≥ 𝑑 ( 𝑝, 𝑝 (cid:48) ) + (cid:104)[− log 𝑝 (cid:48) 𝑝 ] ♭ , log 𝑝 (cid:48) 𝑞 (cid:105) − 𝐹 ( 𝑞 ) , Taking the supremum with respect to 𝑞 on both sides, we conclude the proof. (cid:3) We now establish a result regarding the properness of the conjugate function, thereby generalizing aresult from Bergmann, Herzog, et al., 2021, Lem. 3.4.
Proposition 3.8.
Let 𝐹 : M → R . If 𝐹 ∗ : T ∗ M → R is proper, then 𝐹 is also proper.Proof. Since 𝐹 ∗ is proper by assumption we have dom 𝐹 ∗ ≠ ∅ . Choose some ( 𝑝, 𝜉 ) ∈ dom 𝐹 ∗ . UsingDenition 3.1 we can say that +∞ > 𝐹 ∗ ( 𝑝, 𝜉 ) ≥ (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) for all 𝑞 ∈ M . Since −(−∞) = +∞ , we have that 𝐹 ( 𝑞 ) ≠ −∞ for all 𝑞 ∈ M . Now, we will show that dom 𝐹 ≠ ∅ .Suppose, by contraposition, that 𝐹 ( 𝑞 ) = +∞ for all 𝑞 ∈ M . This would imply that (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) = −∞ for all 𝑞 ∈ M and, consequently, 𝐹 ∗ ( 𝑝, 𝜉 ) = −∞ , which contradicts the fact ( 𝑝, 𝜉 ) ∈ dom 𝐹 ∗ .Therefore, dom 𝐹 ≠ ∅ and proof is complete. (cid:3) Due to the relationship between Denition 3.1 and Bergmann, Herzog, et al., 2021, Def. 3.1 mentioned inRemark 3.3, the proof of the following result follows directly from Lem. 3.7. and Prop. 3.9 of Bergmann,Herzog, et al., 2021. Its proof will therefore be omitted.
Proposition 3.9.
Let
𝐹, 𝐺 : M → R be two proper functions and suppose that 𝛼 ∈ R and 𝜆 > . Thenthe following statements hold. ( 𝑖 ) If 𝐹 ( 𝑞 ) ≤ 𝐺 ( 𝑞 ) for all 𝑞 ∈ M , then 𝐹 ∗ ( 𝑝, 𝜉 ) ≥ 𝐺 ∗ ( 𝑝, 𝜉 ) for all ( 𝑝, 𝜉 ) ∈ T ∗ M . ( 𝑖𝑖 ) If 𝐺 ( 𝑞 ) = 𝐹 ( 𝑞 ) + 𝛼 for all 𝑞 ∈ M , then 𝐺 ∗ ( 𝑝, 𝜉 ) = 𝐹 ∗ ( 𝑝, 𝜉 ) − 𝛼 for all ( 𝑝, 𝜉 ) ∈ T ∗ M . ( 𝑖𝑖𝑖 ) If 𝐺 ( 𝑞 ) = 𝜆𝐹 ( 𝑞 ) for all 𝑞 ∈ M , then 𝐺 ∗ ( 𝑝, 𝜉 ) = 𝜆𝐹 ∗ ( 𝑝, 𝜉𝜆 ) for all ( 𝑝, 𝜉 ) ∈ T ∗ M . ( 𝑖𝑣 ) The Fenchel–Young inequality holds, i. e., for all ( 𝑝, 𝜉 ) ∈ T ∗ M we have 𝐹 ( 𝑞 ) + 𝐹 ∗ ( 𝑝, 𝜉 ) ≥ (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) for all 𝑞 ∈ M . Now we present a result that shows the partial convexity of the Fenchel conjugate w.r.t. the secondargument.
Theorem 3.10.
Let 𝐹 : M → R and 𝑝 ∈ M . Then the function 𝐹 ∗ ( 𝑝, ·) : T ∗ 𝑝 M → R is convex. cbnacbna
Let 𝐹 : M → R and 𝑝 ∈ M . Then the function 𝐹 ∗ ( 𝑝, ·) : T ∗ 𝑝 M → R is convex. cbnacbna page 11 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Proof.
Suppose that 𝜉, 𝜉 (cid:48) ∈ T ∗ 𝑝 M and 𝜆 ∈ [ , ] . Using Denition 3.1 we get 𝐹 ∗ (cid:0) 𝑝, ( − 𝜆 ) 𝜉 + 𝜆𝜉 (cid:48) (cid:1) = sup 𝑞 ∈M (cid:8) (cid:104)( − 𝜆 ) 𝜉 + 𝜆𝜉 (cid:48) , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) = sup 𝑞 ∈M (cid:8) ( − 𝜆 )(cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) + 𝜆 (cid:104) 𝜉 (cid:48) , log 𝑝 𝑞 (cid:105) − ( − 𝜆 ) 𝐹 ( 𝑞 ) − 𝜆𝐹 ( 𝑞 ) (cid:9) = sup 𝑞 ∈M (cid:8) ( − 𝜆 ) (cid:0) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:1) + 𝜆 (cid:0) (cid:104) 𝜉 (cid:48) , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:1) (cid:9) ≤ ( − 𝜆 ) sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) + 𝜆 sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 (cid:48) , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) = ( − 𝜆 ) 𝐹 ∗ ( 𝑝, 𝜉 ) + 𝜆𝐹 ∗ ( 𝑝, 𝜉 (cid:48) ) , which shows the result. (cid:3) Remark 3.11.
Let 𝑝 ∈ M and suppose that 𝐹 ∗ ( 𝑝, ·) : T ∗ 𝑝 M → R is proper. The subdierential of 𝐹 ∗ ( 𝑝, ·) at 𝜉 ∈ T ∗ 𝑝 M , denoted by 𝜕 𝐹 ∗ ( 𝑝, 𝜉 ) , is the set 𝜕 𝐹 ∗ ( 𝑝, 𝜉 ) = (cid:8) 𝑋 ∈ T 𝑝 M (cid:12)(cid:12) 𝐹 ∗ ( 𝑝, 𝜉 (cid:48) ) ≥ 𝐹 ∗ ( 𝑝, 𝜉 ) + (cid:104) 𝜉 (cid:48) − 𝜉 , 𝑋 (cid:105) for all 𝜉 (cid:48) ∈ T ∗ M (cid:9) . In the following statement we give a characterization of this subdierential in terms of the conjugatefunction. This result is a generalization of Theorem 2.4 to the Riemannian context.
Theorem 3.12.
Let 𝐹 : M → R be a proper convex function. Then 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) holds if and only if 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) .Proof. First we consider the case 𝑝 ∈ dom 𝐹 . Suppose that 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) . Hence, using Denition 2.7 wehave (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) ≤ − 𝐹 ( 𝑝 ) for all 𝑞 ∈ M . Taking the supremum with respect to 𝑞 and considering Denition 3.1 we obtain 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) ≤ − 𝐹 ( 𝑝 ) holds. On the other hand, using Denition 3.1 it is easy to see that 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) ≥ (cid:104) 𝜉 , log 𝑝 𝑝 (cid:105) − 𝐹 ( 𝑝 ) = − 𝐹 ( 𝑝 ) . Thus, 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) follows.For the converse, suppose that 𝜉 ∈ T ∗ 𝑝 M is chosen such that 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) holds. Hence, usingDenition 3.1 we have − 𝐹 ( 𝑝 ) = 𝐹 ∗ ( 𝑝, 𝜉 ) ≥ (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) for all 𝑞 ∈ M . Therefore, it follows from Denition 2.7 that 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) holds. cbnacbna
Let 𝐹 : M → R be a proper convex function. Then 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) holds if and only if 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) .Proof. First we consider the case 𝑝 ∈ dom 𝐹 . Suppose that 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) . Hence, using Denition 2.7 wehave (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) ≤ − 𝐹 ( 𝑝 ) for all 𝑞 ∈ M . Taking the supremum with respect to 𝑞 and considering Denition 3.1 we obtain 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) ≤ − 𝐹 ( 𝑝 ) holds. On the other hand, using Denition 3.1 it is easy to see that 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈M (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) ≥ (cid:104) 𝜉 , log 𝑝 𝑝 (cid:105) − 𝐹 ( 𝑝 ) = − 𝐹 ( 𝑝 ) . Thus, 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) follows.For the converse, suppose that 𝜉 ∈ T ∗ 𝑝 M is chosen such that 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) holds. Hence, usingDenition 3.1 we have − 𝐹 ( 𝑝 ) = 𝐹 ∗ ( 𝑝, 𝜉 ) ≥ (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) for all 𝑞 ∈ M . Therefore, it follows from Denition 2.7 that 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) holds. cbnacbna page 12 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds When 𝑝 ∉ dom 𝐹 , then 𝐹 ( 𝑝 ) = ∞ and therefore 𝜕𝐹 ( 𝑝 ) = ∅ since 𝐹 is proper. Suppose that there exists 𝜉 ∈ T ∗ 𝑝 M such that 𝐹 ∗ ( 𝑝, 𝜉 ) = − 𝐹 ( 𝑝 ) holds. Proceeding as above this entails − 𝐹 ( 𝑝 ) = 𝐹 ∗ ( 𝑝, 𝜉 ) ≥(cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) for all 𝑞 ∈ M and therefore 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) , which is a contradiction. This concludes theproof. (cid:3) Remark 3.13.
In case M = R 𝑛 , Theorem 3.12 reads: 𝜉 ∈ 𝜕𝐹 ( 𝑝 ) if and only if − 𝐹 ( 𝑝 ) = 𝐹 ∗ ( 𝑝, 𝜉 ) = 𝐹 ∗ ( , 𝜉 ) − (cid:104) 𝜉 , 𝑝 (cid:105) , where the last equality follows from (3.2) . Since 𝐹 ∗ ( , 𝜉 ) coincides with the classical denition of theFenchel conjugate of 𝐹 , we can indeed conclude that Theorem 3.12 generalizes Theorem 2.4 with X = R 𝑛 tothe Riemannian case. The following result shows that, under certain conditions, a function 𝐹 : M → R is bounded frombelow by a particular continuous function. This function depends on the metric of M and, in theEuclidean case, it is an ane function. A version of this result in Euclidean spaces can be found inZălinescu, 2002, Thm. 2.2.6. Lemma 3.14.
Let 𝐹 : M → R be a proper lsc convex function and 𝑝 ∈ dom 𝐹 . Then there exist 𝑞 ∈ dom 𝐹 , 𝛼 ∈ R and 𝜆 > such that 𝜆 ( log 𝑞 𝑝 , log 𝑞 𝑞 (cid:48) ) − 𝐹 ( 𝑞 (cid:48) ) ≤ 𝛼 for all 𝑞 (cid:48) ∈ dom 𝐹 .
Proof.
First take 𝑠 < 𝐹 ( 𝑝 ) , i. e., ( 𝑝, 𝑠 ) ∉ epi 𝐹 . Since 𝐹 is a proper lsc convex function, applyingCorollary 2.9 we conclude that there exist ( ˆ 𝑝, ˆ 𝑠 ) ∈ epi 𝐹 such that ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑞 (cid:48) ) + ( 𝑠 − ˆ 𝑠 ) ( 𝑟 − ˆ 𝑠 ) ≤ ( 𝑞 (cid:48) , 𝑟 ) ∈ epi 𝐹 . (3.9)Taking ( 𝑞 (cid:48) , 𝑟 ) = ( 𝑝, 𝐹 ( 𝑝 ) + 𝑛 ) , 𝑛 ∈ N , we get ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑝 ) + ( 𝑠 − ˆ 𝑠 ) ( 𝐹 ( 𝑝 ) + 𝑛 − ˆ 𝑠 ) ≤ 𝑛 ∈ N . (3.10)From this 𝑠 − ˆ 𝑠 ≠ 𝑠 = ˆ 𝑠 and, by (3.10), 𝑝 = ˆ 𝑝 follows. Thereforewe would have ( 𝑝, 𝑠 ) = ( ˆ 𝑝, ˆ 𝑠 ) , contradicting the fact ( 𝑝, 𝑠 ) ∉ epi 𝐹 . On the other hand, considering(3.10) with 𝑠 − ˆ 𝑠 > 𝑛 suciently large, we obtain another contradiction. Therefore, we conclude 𝑠 − ˆ 𝑠 <
0. Dividing (3.9) by ˆ 𝑠 − 𝑠 >
0, we have1ˆ 𝑠 − 𝑠 ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑞 (cid:48) ) − 𝑟 ≤ − ˆ 𝑠 for all ( 𝑞 (cid:48) , 𝑟 ) ∈ epi 𝐹 .
Since ( 𝑞 (cid:48) , 𝐹 ( 𝑞 (cid:48) )) ∈ epi 𝐹 for all 𝑞 (cid:48) ∈ dom 𝐹 , it follows that1ˆ 𝑠 − 𝑠 ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑞 (cid:48) ) − 𝐹 ( 𝑞 (cid:48) ) ≤ − ˆ 𝑠 for all 𝑞 (cid:48) ∈ dom 𝐹 .
To nalize the proof choose 𝑞 = ˆ 𝑝 , 𝛼 = − ˆ 𝑠 and 𝜆 = /( ˆ 𝑠 − 𝑠 ) . (cid:3) cbnacbna
To nalize the proof choose 𝑞 = ˆ 𝑝 , 𝛼 = − ˆ 𝑠 and 𝜆 = /( ˆ 𝑠 − 𝑠 ) . (cid:3) cbnacbna page 13 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds The following result shows that our denition of Fenchel conjugate on M allows us to obtain anextension of the second part of Theorem 2.5 to the Riemannian context. Theorem 3.15.
Let 𝐹 : M → R be a proper lsc convex function. Then 𝐹 ∗ is proper.Proof. Fix ( 𝑝, 𝜉 ) ∈ T ∗ M and choose some 𝑝 (cid:48) ∈ dom 𝐹 . Using Remark 3.2, we have 𝐹 ∗ ( 𝑝, 𝜉 ) = sup 𝑞 ∈ dom 𝐹 (cid:8) (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) − 𝐹 ( 𝑞 ) (cid:9) ≥ (cid:104) 𝜉 , log 𝑝 𝑝 (cid:48) (cid:105) − 𝐹 ( 𝑝 (cid:48) ) > −∞ . On the other hand, Lemma 3.14 guarantees that there are 𝑞 ∈ dom 𝐹 , 𝛼 ∈ R and 𝜆 > 𝜆 ( log 𝑞 𝑝 (cid:48) , log 𝑞 𝑞 (cid:48) ) − 𝐹 ( 𝑞 (cid:48) ) ≤ 𝛼 for all 𝑞 (cid:48) ∈ dom 𝐹 .
Using (2.2) and taking the supremum with respect to 𝑞 (cid:48) , we can conclude that 𝐹 ∗ (cid:0) 𝑞, [ 𝜆 log 𝑞 𝑝 (cid:48) ] ♭ (cid:1) ≤ 𝛼 < +∞ holds. This shows dom 𝐹 ∗ ≠ ∅ , which completes the proof. (cid:3) We now dene the Fenchel biconjugate on Hadamard manifolds.
Denition 3.16.
Let 𝐹 : M → R . The Fenchel biconjugate of 𝐹 is the function 𝐹 ∗∗ : M → R dened by 𝐹 ∗∗ ( 𝑝 ) (cid:66) sup ( 𝑞,𝜉 ) ∈T ∗ M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) − 𝐹 ∗ ( 𝑞, 𝜉 ) (cid:9) for 𝑝 ∈ M . Similarly as it was the case for 𝐹 ∗ , our denition diers from our previous denition of the biconjugatein Bergmann, Herzog, et al., 2021, Def. 3.5. In particular, 𝐹 ∗∗ does not depend on a base point. Thefollowing remark shows that the above denition is a natural extension of (2.1) to the Riemanniancontext. Remark 3.17.
Let 𝐹 : R 𝑛 → R . The Denition 3.16 with M equal to the Euclidean space R 𝑛 becomes 𝐹 ∗∗ ( 𝑝 ) = sup ( 𝑞,𝜉 ) ∈ R 𝑛 × R 𝑛 (cid:8) (cid:104) 𝜉 , 𝑝 − 𝑞 (cid:105) − 𝐹 ∗ ( 𝑞, 𝜉 ) (cid:9) for 𝑝 ∈ R 𝑛 . Since Remark 3.3, Equation (3.2) states that 𝐹 ∗ ( 𝑞, 𝜉 ) = 𝐹 ∗ ( , 𝜉 ) − (cid:104) 𝜉 , 𝑞 (cid:105) for all ( 𝑞, 𝜉 ) ∈ R 𝑛 × R 𝑛 , it followsthat 𝐹 ∗∗ ( 𝑝 ) = sup 𝜉 ∈ R 𝑛 (cid:8) (cid:104) 𝜉 , 𝑝 (cid:105) − 𝐹 ∗ ( , 𝜉 ) (cid:9) for 𝑝 ∈ R 𝑛 . Taking into account that 𝐹 ∗ ( , 𝜉 ) coincides with the classical denition of Fenchel conjugate on R 𝑛 , wecan conclude that Denition 3.16 generalizes the classical denition of the Fenchel biconjugate from theEuclidean space to Hadamard manifolds. cbnacbna
Let 𝐹 : R 𝑛 → R . The Denition 3.16 with M equal to the Euclidean space R 𝑛 becomes 𝐹 ∗∗ ( 𝑝 ) = sup ( 𝑞,𝜉 ) ∈ R 𝑛 × R 𝑛 (cid:8) (cid:104) 𝜉 , 𝑝 − 𝑞 (cid:105) − 𝐹 ∗ ( 𝑞, 𝜉 ) (cid:9) for 𝑝 ∈ R 𝑛 . Since Remark 3.3, Equation (3.2) states that 𝐹 ∗ ( 𝑞, 𝜉 ) = 𝐹 ∗ ( , 𝜉 ) − (cid:104) 𝜉 , 𝑞 (cid:105) for all ( 𝑞, 𝜉 ) ∈ R 𝑛 × R 𝑛 , it followsthat 𝐹 ∗∗ ( 𝑝 ) = sup 𝜉 ∈ R 𝑛 (cid:8) (cid:104) 𝜉 , 𝑝 (cid:105) − 𝐹 ∗ ( , 𝜉 ) (cid:9) for 𝑝 ∈ R 𝑛 . Taking into account that 𝐹 ∗ ( , 𝜉 ) coincides with the classical denition of Fenchel conjugate on R 𝑛 , wecan conclude that Denition 3.16 generalizes the classical denition of the Fenchel biconjugate from theEuclidean space to Hadamard manifolds. cbnacbna page 14 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds The Fenchel biconjugate function is always a lower bound on the original function, as the followingresult states, which generalizes Bauschke, Combettes, 2011, Prop. 13.14.
Proposition 3.18.
Let 𝐹 : M → R . Then 𝐹 ∗∗ ( 𝑝 ) ≤ 𝐹 ( 𝑝 ) holds for all 𝑝 ∈ M .Proof. Applying Denition 3.16 and Denition 3.1, we have 𝐹 ∗∗ ( 𝑝 ) = sup ( 𝑞,𝜉 ) ∈T ∗ M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) − 𝐹 ∗ ( 𝑞, 𝜉 ) (cid:9) , = sup ( 𝑞,𝜉 ) ∈T ∗ M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) − sup 𝑞 (cid:48) ∈M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑞 (cid:48) (cid:105) − 𝐹 ( 𝑞 (cid:48) ) (cid:9)(cid:9) , = sup ( 𝑞,𝜉 ) ∈T ∗ M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) + inf 𝑞 (cid:48) ∈M (cid:8) −(cid:104) 𝜉 , log 𝑞 𝑞 (cid:48) (cid:105) + 𝐹 ( 𝑞 (cid:48) ) (cid:9)(cid:9) , ≤ sup ( 𝑞,𝜉 ) ∈T ∗ M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) − (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) + 𝐹 ( 𝑝 ) (cid:9) , = 𝐹 ( 𝑝 ) for any 𝑝 ∈ M . (cid:3) The following result is a version of the famous Fenchel–Moreau theorem in the Riemannian case,compare Theorem 2.5.
Theorem 3.19.
Let 𝐹 : M → R be a proper lsc convex function. Then 𝐹 ∗∗ = 𝐹 holds.Proof. Let 𝑝 ∈ M be arbitrary. Choose some 𝑠 ∈ R such that 𝑠 < 𝐹 ( 𝑝 ) holds, i. e., ( 𝑝, 𝑠 ) ∉ epi 𝐹 . Since 𝐹 is a proper lsc convex function, we can apply Corollary 2.9 to conclude that there exists ( ˆ 𝑝, ˆ 𝑠 ) ∈ epi 𝐹 such that ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑞 ) + ( 𝑠 − ˆ 𝑠 ) ( 𝑟 − ˆ 𝑠 ) ≤ ( 𝑞, 𝑟 ) ∈ epi 𝐹 . (3.11)Considering (3.11) with ( 𝑞, 𝑟 ) = ( ˆ 𝑝, 𝐹 ( ˆ 𝑝 ) + 𝑛 ) , 𝑛 ∈ N , we obtain ( 𝑠 − ˆ 𝑠 ) ( 𝐹 ( ˆ 𝑝 ) + 𝑛 − ˆ 𝑠 ) ≤ 𝑛 ∈ N . Since ˆ 𝑝 ∈ dom 𝐹 holds, the assumption 𝑠 − ˆ 𝑠 > 𝑛 suciently large.Therefore, we must have 𝑠 − ˆ 𝑠 ≤
0. With this in mind, we will prove 𝑠 ≤ 𝐹 ∗∗ ( 𝑝 ) .First, let us assume 𝑠 − ˆ 𝑠 <
0. Dividing (3.11) by ˆ 𝑠 − 𝑠 > 𝑠 − 𝑠 ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑝 (cid:48) ) − 𝑟 ≤ − ˆ 𝑠 < − 𝑠 for all ( 𝑝 (cid:48) , 𝑟 ) ∈ epi 𝐹 .
Using (2.2) and the expression above with 𝑝 (cid:48) ∈ dom 𝐹 and 𝑟 = 𝐹 ( 𝑝 (cid:48) ) , we have (cid:104)[( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ , log ˆ 𝑝 𝑝 (cid:48) (cid:105) − 𝐹 ( 𝑝 (cid:48) ) = 𝑠 − 𝑠 ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑝 (cid:48) ) − 𝐹 ( 𝑝 (cid:48) ) < − 𝑠 for all 𝑝 (cid:48) ∈ dom 𝐹 .
Taking the supremum with respect to 𝑝 (cid:48) ∈ dom 𝐹 and considering Remark 3.2, it follows that 𝐹 ∗ (cid:0) ˆ 𝑝, [( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ (cid:1) ≤ − 𝑠 . Taking into account that (cid:104)[( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ , log ˆ 𝑝 𝑝 (cid:105) ≥
0, the lastinequality and Denition 3.16 yield 𝑠 ≤ (cid:104)[( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ , log ˆ 𝑝 𝑝 (cid:105) − 𝐹 ∗ (cid:0) ˆ 𝑝, [( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ (cid:1) ≤ 𝐹 ∗∗ ( 𝑝 ) . cbnacbna
0, the lastinequality and Denition 3.16 yield 𝑠 ≤ (cid:104)[( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ , log ˆ 𝑝 𝑝 (cid:105) − 𝐹 ∗ (cid:0) ˆ 𝑝, [( log ˆ 𝑝 𝑝 )/( ˆ 𝑠 − 𝑠 )] ♭ (cid:1) ≤ 𝐹 ∗∗ ( 𝑝 ) . cbnacbna page 15 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Now, let us prove that 𝑠 ≤ 𝐹 ∗∗ ( 𝑝 ) also holds when we assume 𝑠 − ˆ 𝑠 =
0. In this case, (3.11) becomes ( log ˆ 𝑝 𝑝 , log ˆ 𝑝 𝑞 ) ≤ 𝑞 ∈ dom 𝐹 . (3.12)Using Lemma 3.14 for ˆ 𝑝 ∈ dom 𝐹 , there exist 𝑞 (cid:48) ∈ dom 𝐹 , 𝛼 ∈ R and 𝜆 > 𝜆 ( log 𝑞 (cid:48) ˆ 𝑝 , log 𝑞 (cid:48) 𝑞 ) − 𝐹 ( 𝑞 ) ≤ 𝛼 for all 𝑞 ∈ dom 𝐹 .
On the other hand, it is easy to see that (3.12) is equivalent to the inequality 𝜆 ( log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑞 ) + 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑞 ) ≤ 𝑞 ∈ dom 𝐹, 𝜎 > . Adding the last two inequalities we get 𝜆 ( log 𝑞 (cid:48) ˆ 𝑝 , log 𝑞 (cid:48) 𝑞 ) + 𝜆 ( log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑞 )+ 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑞 ) − 𝐹 ( 𝑞 ) ≤ 𝛼 for all 𝑞 ∈ dom 𝐹, 𝜎 > . (3.13)Using (2.6) for the geodesic triangle Δ ( ˆ 𝑝, 𝑞 (cid:48) , 𝑞 ) for 𝑞 ∈ M , we can conclude that 0 ≤ ( log 𝑞 (cid:48) ˆ 𝑝 , log 𝑞 (cid:48) 𝑞 ) +( log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑞 ) holds for all 𝑞 ∈ dom 𝐹 . Thus, (3.13) yields 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑞 ) − 𝐹 ( 𝑞 ) ≤ 𝛼 for all 𝑞 ∈ dom 𝐹, 𝜎 > . Considering (2.2) and taking the supremum over 𝑞 ∈ M , we get 𝐹 ∗ (cid:0) ˆ 𝑝, [ 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) )] ♭ (cid:1) ≤ 𝛼 for all 𝜎 >
0. Therefore, 𝜆 𝜎 𝑑 ( 𝑝, ˆ 𝑝 ) − 𝜆 ( log ˆ 𝑝 𝑞 (cid:48) , log ˆ 𝑝 𝑝 ) − 𝛼 = ( 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) ) , log ˆ 𝑝 𝑝 ) − 𝛼 = (cid:104)[ 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) )] ♭ , log ˆ 𝑝 𝑝 (cid:105) − 𝛼 ≤ (cid:104)[ 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) )] ♭ , log ˆ 𝑝 𝑝 (cid:105) − 𝐹 ∗ (cid:0) ˆ 𝑝, [ 𝜆 ( 𝜎 log ˆ 𝑝 𝑝 − log ˆ 𝑝 𝑞 (cid:48) )] ♭ (cid:1) ≤ sup ( 𝑞,𝜉 ) ∈T ∗ M (cid:8) (cid:104) 𝜉 , log 𝑞 𝑝 (cid:105) − 𝐹 ∗ ( 𝑞, 𝜉 ) (cid:9) = 𝐹 ∗∗ ( 𝑝 ) , for all 𝜎 >
0. As we are analyzing the case 𝑠 − ˆ 𝑠 =
0, we can conclude that 𝑝 ≠ ˆ 𝑝 must hold, otherwisewe would have ( 𝑝, 𝑠 ) = ( ˆ 𝑝, ˆ 𝑠 ) , contradicting the fact ( 𝑝, 𝑠 ) ∉ epi 𝐹 . Thus, taking 𝜎 suciently large, weget 𝐹 ∗∗ ( 𝑝 ) = +∞ and thus 𝑠 ≤ 𝐹 ∗∗ ( 𝑝 ) holds in this case as well.We have thus proved 𝑠 ≤ 𝐹 ∗∗ ( 𝑝 ) in all cases. Since 𝑠 < 𝐹 ( 𝑝 ) was arbitrary, we get 𝐹 ( 𝑝 ) ≤ 𝐹 ∗∗ ( 𝑝 ) . Theconclusion of the proof now follows from Proposition 3.18. (cid:3) Throughout this section, we show that the structure of T ∗ M enables us to obtain a theory of separationof convex sets on a Hadamard manifold M . To see this theory on normed vector space we refer thereader, e. g., to Brezis, 2011, Ch. 1. We begin by introducing a concept that generalizes the denition ofan ane hyperplane to the Riemannian context. cbnacbna
0, we can conclude that 𝑝 ≠ ˆ 𝑝 must hold, otherwisewe would have ( 𝑝, 𝑠 ) = ( ˆ 𝑝, ˆ 𝑠 ) , contradicting the fact ( 𝑝, 𝑠 ) ∉ epi 𝐹 . Thus, taking 𝜎 suciently large, weget 𝐹 ∗∗ ( 𝑝 ) = +∞ and thus 𝑠 ≤ 𝐹 ∗∗ ( 𝑝 ) holds in this case as well.We have thus proved 𝑠 ≤ 𝐹 ∗∗ ( 𝑝 ) in all cases. Since 𝑠 < 𝐹 ( 𝑝 ) was arbitrary, we get 𝐹 ( 𝑝 ) ≤ 𝐹 ∗∗ ( 𝑝 ) . Theconclusion of the proof now follows from Proposition 3.18. (cid:3) Throughout this section, we show that the structure of T ∗ M enables us to obtain a theory of separationof convex sets on a Hadamard manifold M . To see this theory on normed vector space we refer thereader, e. g., to Brezis, 2011, Ch. 1. We begin by introducing a concept that generalizes the denition ofan ane hyperplane to the Riemannian context. cbnacbna page 16 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Denition 4.1. An ane hypersurface of M is a set H ⊂ M of the form H = H ( 𝑝, 𝜉, 𝛼 ) (cid:66) { 𝑞 ∈ M | (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) = 𝛼 } , where ( 𝑝, 𝜉 ) ∈ T ∗ M and 𝛼 ∈ R are given with 𝜉 ≠ . Remark 4.2.
Consider the equivalence relation ∼ dened in (3.4) . Note that H ( 𝑝, 𝜉, 𝛼 ) = H ( 𝑝 (cid:48) , 𝜉 (cid:48) , 𝛼 ) holds for all ( 𝑝 (cid:48) , 𝜉 (cid:48) ) ∼ ( 𝑝, 𝜉 ) . Denition 4.3.
Let A and B be two subsets of M . We say that the hypersurface H ( 𝑝, 𝜉, 𝛼 ) separates A and B if (cid:104) 𝜉 , log 𝑝 𝑞 (cid:105) ≤ 𝛼 ≤ (cid:104) 𝜉 , log 𝑝 𝑞 (cid:48) (cid:105) holds for all 𝑞 ∈ A , 𝑞 (cid:48) ∈ B . (4.1) We say that
H ( 𝑝, 𝜉, 𝛼 ) strictly separates A and B when both inequalities above are strict. Geometrically speaking, (4.1) means that A lies in one of the “half-manifolds” determined by H , and B lies in the other.Let A be a subset of M . It is well known that the function 𝜙 : M → R dened by 𝜙 ( 𝑝 ) = 𝑑 ( 𝑝, A) (cid:66) inf 𝑞 ∈A 𝑑 ( 𝑝, 𝑞 ) (4.2)is continuous on M . This property will be used in the proof of the following result, which extends theclassical strict separation theorem to the Riemannian setting. Theorem 4.4.
Let
A ⊂ M and
B ⊂ M be two nonempty convex subsets such that
A ∩ B = ∅ , A isclosed and B is compact. Then there exists a hypersurface which strictly separates A and B .Proof. Throughout the proof, points in B will be marked by a prime. Since 𝜙 dened as in (4.2) iscontinuous and B is compact, the problem of minimizing 𝜙 over B possesses at least one globalsolution. We denote one such solution by ˆ 𝑞 (cid:48) ∈ B , i. e., 𝑑 ( ˆ 𝑞 (cid:48) , A) ≤ 𝑑 ( 𝑞, A) holds for all 𝑞 ∈ B .As A is convex and closed, the projection map proj A : M → A is well dened. Hence, settingˆ 𝑞 (cid:66) proj A ( ˆ 𝑞 (cid:48) ) , we have 𝑑 ( 𝑞 (cid:48) , ˆ 𝑞 ) ≥ 𝑑 ( 𝑞 (cid:48) , A) ≥ min 𝑞 (cid:48) ∈B 𝑑 ( 𝑞 (cid:48) , A) = 𝜙 ( ˆ 𝑞 (cid:48) ) = 𝑑 ( ˆ 𝑞 (cid:48) , A) = 𝑑 ( ˆ 𝑞 (cid:48) , proj A ( ˆ 𝑞 (cid:48) )) = 𝑑 ( ˆ 𝑞 (cid:48) , ˆ 𝑞 ) for all 𝑞 (cid:48) ∈ B , which means that ˆ 𝑞 (cid:48) = proj B ( ˆ 𝑞 ) . Taking into account A ∩ B = ∅ we deduce ˆ 𝑞 ≠ ˆ 𝑞 (cid:48) .Let us dene 𝑝 to be the midpoint of the geodesic segment connecting ˆ 𝑞 to ˆ 𝑞 (cid:48) . Then it is easy to seethat we have 𝑑 ( ˆ 𝑞, 𝑝 ) = 𝑑 ( ˆ 𝑞 (cid:48) , 𝑝 ) = 𝑑 ( ˆ 𝑞, ˆ 𝑞 (cid:48) ) > . (4.3) cbnacbna
A ∩ B = ∅ , A isclosed and B is compact. Then there exists a hypersurface which strictly separates A and B .Proof. Throughout the proof, points in B will be marked by a prime. Since 𝜙 dened as in (4.2) iscontinuous and B is compact, the problem of minimizing 𝜙 over B possesses at least one globalsolution. We denote one such solution by ˆ 𝑞 (cid:48) ∈ B , i. e., 𝑑 ( ˆ 𝑞 (cid:48) , A) ≤ 𝑑 ( 𝑞, A) holds for all 𝑞 ∈ B .As A is convex and closed, the projection map proj A : M → A is well dened. Hence, settingˆ 𝑞 (cid:66) proj A ( ˆ 𝑞 (cid:48) ) , we have 𝑑 ( 𝑞 (cid:48) , ˆ 𝑞 ) ≥ 𝑑 ( 𝑞 (cid:48) , A) ≥ min 𝑞 (cid:48) ∈B 𝑑 ( 𝑞 (cid:48) , A) = 𝜙 ( ˆ 𝑞 (cid:48) ) = 𝑑 ( ˆ 𝑞 (cid:48) , A) = 𝑑 ( ˆ 𝑞 (cid:48) , proj A ( ˆ 𝑞 (cid:48) )) = 𝑑 ( ˆ 𝑞 (cid:48) , ˆ 𝑞 ) for all 𝑞 (cid:48) ∈ B , which means that ˆ 𝑞 (cid:48) = proj B ( ˆ 𝑞 ) . Taking into account A ∩ B = ∅ we deduce ˆ 𝑞 ≠ ˆ 𝑞 (cid:48) .Let us dene 𝑝 to be the midpoint of the geodesic segment connecting ˆ 𝑞 to ˆ 𝑞 (cid:48) . Then it is easy to seethat we have 𝑑 ( ˆ 𝑞, 𝑝 ) = 𝑑 ( ˆ 𝑞 (cid:48) , 𝑝 ) = 𝑑 ( ˆ 𝑞, ˆ 𝑞 (cid:48) ) > . (4.3) cbnacbna page 17 of 20. Bergmann, R. Herzog and M. Silva Louzeiro Fenchel Duality on Hadamard Manifolds Next we prove proj A ( 𝑝 ) = ˆ 𝑞 and proj B ( 𝑝 ) = ˆ 𝑞 (cid:48) . Suppose by contradiction that proj A ( 𝑝 ) ≠ ˆ 𝑞 and consider the geodesic triangle Δ ( ˆ 𝑞, 𝑝, proj A ( 𝑝 )) . Since log ˆ 𝑞 𝑝 = log ˆ 𝑞 ˆ 𝑞 (cid:48) and ˆ 𝑞 = proj A ( ˆ 𝑞 (cid:48) ) ,Theorem 2.8 guarantees ( log ˆ 𝑞 𝑝 , log ˆ 𝑞 𝑞 ) = ( log ˆ 𝑞 ˆ 𝑞 (cid:48) , log ˆ 𝑞 𝑞 ) ≤ 𝑞 ∈ A , (4.4) ( log proj A ( 𝑝 ) 𝑝 , log proj A ( 𝑝 ) 𝑞 ) ≤ 𝑞 ∈ A . (4.5)Taking (4.4) with 𝑞 = proj A ( 𝑝 ) and (4.5) with 𝑞 = ˆ 𝑞 we get ( log ˆ 𝑞 𝑝 , log ˆ 𝑞 proj A ( 𝑝 )) + ( log proj A ( 𝑝 ) 𝑝 , log proj A ( 𝑝 ) ˆ 𝑞 ) ≤ , which contradicts (2.6) for Δ ( ˆ 𝑞, 𝑝, proj A ( 𝑝 )) . Thus, we can conclude that proj A ( 𝑝 ) = ˆ 𝑞 holds. Actinganalogously with the geodesic triangle Δ ( ˆ 𝑞 (cid:48) , 𝑝, proj B ( 𝑝 )) , we can also conclude that proj B ( 𝑝 ) = ˆ 𝑞 (cid:48) .Consider the geodesic triangle Δ ( ˆ 𝑞, 𝑝, 𝑞 ) , 𝑞 ∈ A . Since proj A ( 𝑝 ) = ˆ 𝑞 , Theorem 2.8 and (2.6) guaranteethat −( log ˆ 𝑞 𝑝 , log ˆ 𝑞 𝑞 ) ≥ 𝑞 ∈ A , ( log ˆ 𝑞 𝑝 , log ˆ 𝑞 𝑞 ) + ( log 𝑝 ˆ 𝑞 , log 𝑝 𝑞 ) ≥ 𝑑 ( 𝑝, ˆ 𝑞 ) for all 𝑞 ∈ A . Adding the two inequalities above and using (4.3) we can deduce that ( log 𝑝 ˆ 𝑞 , log 𝑝 𝑞 ) ≥ 𝑑 ( 𝑝, ˆ 𝑞 ) > 𝑞 ∈ A . (4.6)Similarly, considering the geodesic triangle Δ ( ˜ 𝑞 (cid:48) , 𝑝, 𝑞 (cid:48) ) , 𝑞 (cid:48) ∈ B , and taking into account that proj B ( 𝑝 ) = ˆ 𝑞 (cid:48) , we can also say that ( log 𝑝 ˆ 𝑞 (cid:48) , log 𝑝 𝑞 (cid:48) ) ≥ 𝑑 ( 𝑝, ˆ 𝑞 (cid:48) ) > 𝑞 (cid:48) ∈ B . Since log 𝑝 ˆ 𝑞 (cid:48) = − log 𝑝 ˆ 𝑞 , the last inequality implies ( log 𝑝 ˆ 𝑞 , log 𝑝 𝑞 (cid:48) ) < 𝑞 (cid:48) ∈ B . Hence, using(4.6) and (2.2) we can conclude that the hypersurface H (cid:0) 𝑝, [ log 𝑝 ˆ 𝑞 ] ♭ , (cid:1) strictly separates A and B . (cid:3) In this paper we introduced a new denition of the Fenchel conjugate for functions dened onHadamard manifolds. In contrast to previous denitions, it is independent of the choice of a base point.Our concept generalizes the Fenchel conjugate in the Euclidean case, and essential properties carryover. As a next step we plan to investigate how to leverage the new concept algorithmically. Moreover,we expect that a weaker version of the separation theorem can be shown, which merely requires A and B to be convex and closed. References
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