aa r X i v : . [ m a t h . F A ] F e b A primal representation of theMonge-Kantorovich norm
D´avid Terj´ekAlfr´ed R´enyi Institute of MathematicsBudapest, Hungary [email protected]
February 25, 2021
Abstract
In this note, following [Chitescu et al., 2014], we show that the Monge-Kantorovich norm on the vector space of countably additive measures on acompact metric space has a primal representation analogous to the Haninnorm, meaning that similarly to the Hanin norm, the Monge-Kantorovichnorm can be seen as an extension of the Kantorovich-Rubinstein normfrom the vector subspace of zero-charge measures, implying a numberof novel results, such as the equivalence of the Monge-Kantorovich andHanin norms.
As presented in [Chitescu et al., 2014] and summarized in [Cobza¸s et al., 2019],the vector space cabv ( X ) of countably additive measures with bounded varia-tion on a compact metric space ( X, d ) can be normed in at least two distinctways to induce the topology of weak convergence of measures (also called weak- ∗ convergence of measures) on subsets bounded with respect to the total variationnorm. One ( cabv ( X ) , k . k MK ) is by the Monge-Kantorovich norm (also calledthe Kantorovich-Rubinstein norm) k µ k MK = sup f ∈ Lip ( X ) , k f k sum ≤ (cid:26)Z f dµ (cid:27) , (1)with ( Lip ( X ) , k . k sum ) being the vector space of functions f : X → R for whichthe Lipschitz seminorm k f k L = sup x,y ∈ X,x = y (cid:26) | f ( x ) − f ( y ) | d ( x, y ) (cid:27) (2)is finite, normed by the sum norm, which is defined as k f k sum = k f k L + k f k ∞ . (3)1he other ( cabv ( X ) , k . k H ) is by the Hanin norm k µ k H = inf ν ∈ cabv ( X, {k ν k KR + k µ − ν k T V } , (4)where k . k T V is the total variation norm, and ( cabv ( X, , k . k KR ) is the vec-tor subspace cabv ( X,
0) = { µ ∈ cabv ( X ) : µ ( X ) = 0 } of zero-charge mea-sures normed by the Kantorovich-Rubinstein norm (also called the modifiedKantorovich-Rubinstein norm) k µ k KR = sup f ∈ Lip ( X,x ) , k f k L ≤ (cid:26)Z f dµ (cid:27) , (5)with ( Lip ( X, x ) , k . k L ) being the vector subspace Lip ( X, x ) = { f ∈ Lip ( X ) : f ( x ) = 0 } of Lipschitz continuous functions vanishing at an arbitrary fixedpoint x ∈ X , normed by the Lipschitz seminorm.The topological dual of ( cabv ( X, , k . k KR ) and ( Lip ( X, x ) , k . k L ) are iso-metrically isomorphic. The topological dual of ( cabv ( X ) , k . k H ) and ( Lip ( X ) , k . k max )are isometrically isomorphic as well, with the max norm defined as k f k max = max {k f k L , k f k ∞ } , (6)leading to the dual representation k µ k H = sup f ∈ Lip ( X ) , k f k max ≤ (cid:26)Z f dµ (cid:27) . (7)Inspired by the similarity of the dual representation of the Hanin norm andthe definition of the Monge-Kantorovich norm, we prove the following theorem,showing that analogously to the Hanin norm, the Monge-Kantorovich norm canbe seen as an extension of the Kantorovich-Rubinstein norm from the subspace cabv ( X,
0) to the whole space cabv ( X ), leading to a number of consequences. Theorem.
The Monge-Kantorovich norm has the primal representation k µ k MK = inf ν ∈ cabv ( X, { max {k ν k KR , k µ − ν k T V }} , (8) the supremum in the dual representation is achieved as ∃ f ∈ Lip ( X ) : k f k sum =1 ∧ k µ k MK = R f dµ , the subset of measures in cabv ( X ) with finite supportis dense in ( cabv ( X ) , k . k MK ) , the topological dual of ( cabv ( X ) , k . k MK ) and ( Lip ( X ) , k . k sum ) are isometrically isomorphic, and the norms k . k MK and k . k H are equivalent as k µ k MK ≤ k µ k H ≤ k µ k MK for ∀ µ ∈ cabv ( X ) . To obtain the proof, we will apply techniques of convex analysis cited from[Zalinescu, 2002]. We start with a number of propositions.
Proposition 1.
Given λ ∈ [0 , , the mapping F λ : ( cabv ( X, k . k KR ) → R defined as F λ ( ν ) = λ k ν k KR (9) is proper, convex and continuous, and its convex conjugate F ∗ λ : ( Lip ( X, x ) , k . k L ) → R is the indicator F ∗ λ ( f ) = i { f ∈ Lip ( X,x ): k f k L ≤ λ } ( f ) . (10)2 roof. By [Zalinescu, 2002, Corollary 2.4.16],( ν → k ν k KR ) ∗ = ( f → i { f ∈ Lip ( X,x ): k f k L ≤ } ( f )) . (11)By [Zalinescu, 2002, Theorem 2.3.1(v)],( ν → λ k ν k KR ) ∗ = ( f → λi { f ∈ Lip ( X,x ): k f k L ≤ } ( λ − f )) , (12)which is equivalent to the proposed conjugate relation. The mapping is clearlyproper, convex and continuous by being the constant multiple of a norm with apositive multiplier. Proposition 2.
Given λ ∈ [0 , and µ ∈ cabv ( X ) , the mapping G λ,µ : ( cabv ( X ) , k . k H ) → R defined as G λ,µ ( ν ) = (1 − λ ) k µ − ν k T V (13) is proper, convex and lower semicontinuous, and its convex conjugate G ∗ λ,µ :( Lip ( X ) , k . k max ) → R is G ∗ λ ( f ) = i { f ∈ Lip ( X ): k f k ∞ ≤ − λ } ( f ) − Z f dµ. (14) Proof.
By [Cobza¸s et al., 2019, Theorem 8.4.10], the level sets of the mapping( ν → k ν k T V ) are compact with respect to topology of the weak convergence ofmeasures, hence compact with respect to the topology induced by k . k H as well by[Cobza¸s et al., 2019, Theorem 8.5.7]. This implies that the level sets are closedin ( cabv ( X ) , k . k H ), hence the mapping is lower semicontinuous, and clearlyproper and convex as well. It is also sublinear, so that by [Zalinescu, 2002,Theorem 2.4.14(i)] one has the conjugate relation( ν → k ν k T V ) ∗ = i ∂ ( ν →k ν k TV )(0) , (15)where by definition the subdifferential at 0 is ∂ ( ν → k ν k T V )(0) = { f ∈ Lip ( X ) | ∀ µ ∈ cabv ( X ) : Z f dµ ≤ k µ k T V } . (16)Since X is compact, any f ∈ Lip ( X ) achieves its minimum and maximum,hence ∃ x ∈ X : | f ( x ) | = k f k ∞ , (17)implying thatmax (cid:26)Z f dµ : µ ∈ cabv ( X ) , k µ k T V = ξ (cid:27) = Z f d ( ± ξδ x ) = ξ k f k ∞ (18)with δ x being the Dirac measure at x , and the sign of ξ is opposite to that of f ( x ). It follows that ∃ µ ∈ cabv ( X ) , k µ k T V = ξ : Z f dµ > k µ k T V ⇐⇒ k f k ∞ > , (19)3hich is true for any ξ ≥
0, implying that ∂ ( ν → k ν k T V )(0) = { f ∈ Lip ( X ) : k f k ∞ ≤ } , (20)leading to the conjugate relation( ν → k ν k T V ) ∗ = i { f ∈ Lip ( X ): k f k ∞ ≤ } . (21)By [Zalinescu, 2002, Theorem 2.3.1(v)],( ν → k − ν k T V ) ∗ = ( f → i { f ∈ Lip ( X ): k f k ∞ ≤ } ( − f )) , (22)where ( − f ) can clearly be replaced by ( f ). By [Zalinescu, 2002, Theorem 2.3.1(vi)],( ν → k µ − ν k T V ) ∗ = (cid:18) f → i { f ∈ Lip ( X ): k f k ∞ ≤ } ( f ) − Z f dµ (cid:19) . (23)By [Zalinescu, 2002, Theorem 2.3.1(v)],( ν → (1 − λ ) k µ − ν k T V ) ∗ = (cid:0) f → (1 − λ ) i { f ∈ Lip ( X ): k f k ∞ ≤ } ((1 − λ ) − f ) − (1 − λ ) Z (1 − λ ) − f dµ (cid:19) , (24)which is clearly equivalent to the proposition. Proposition 3.
Given µ ∈ cabv ( X ) , the mapping H µ : ( cabv ( X, , k . k KR ) × ( cabv ( X ) , k . k H ) → R defined as H µ ( ν , ν ) = max {k ν k KR , k µ − ν k T V } (25) is proper, convex and lower semicontinuous, and its convex conjugate H ∗ µ :( Lip ( X, x ) , k . k L ) × ( Lip ( X ) , k . k max ) → R is H ∗ µ ( f , f ) = min λ ∈ [0 , (cid:26) i { f ∈ Lip ( X,x ): k f k L ≤ λ } ( f )+ i { f ∈ Lip ( X ): k f k ∞ ≤ − λ } ( f ) − Z f dµ (cid:27) . (26) Proof.
By [Zalinescu, 2002, Corollary 2.8.12], the conjugate relation(( ν , ν ) → max { λ − F λ ( ν ) , (1 − λ ) − G λ,µ ( ν ) } ) ∗ = (( f , f ) → min λ ∈ [0 , { F ∗ λ ( f ) + G ∗ λ,µ ( f ) } ) (27)holds, which together with the previous propositions gives the claimed con-jugate relation. By [Zalinescu, 2002, Theorem 2.1.3(vii)], H µ is proper andconvex. Clearly the mappings ((( ν , ν )) → λ − F λ ( ν )) and ((( ν , ν )) → (1 − λ ) − G λ,µ ( ν )) are lower semicontinuous, hence H µ as well by being theirpointwise maximum. 4 roposition 4. The mapping ( ν → ν ) : ( cabv ( X, , k . k KR → ( cabv ( X ) , k . k H ) is linear and continuous, and its adjoint ( ν → ν ) ∗ is ( f → f − f ( x )) :( Lip ( X ) , k . k max ) : ( Lip ( X, x ) , k . k L ) .Proof. It is clear that k ν k H ≤ k ν k KR , so the linear operator ( ν → ν ) isbounded, hence continuous. For ∀ ν ∈ cabv ( X, , f ∈ Lip ( X ) it holds that R ( f − f ( x )) dν = R f dν − f ( x ) ν ( X ) = R f dν , proving the adjoint relation.We are now ready to prove the theorem. Proof of Theorem .
First we show that the condition [Zalinescu, 2002, Theo-rem 2.8.1(iii)] holds for the mapping H µ . Since H µ is finite everywhere, one hasdom H µ = cabv ( X, × cabv ( X ). For any ν ∈ cabv ( X, H µ ( ν , · ) to any closed subset is lower semicontinuous. For any r > { ν ∈ cabv ( X ) : k ν k T V ≤ r } is complete withrespect to any metric metrizing the topology of the weak convergence of mea-sures by [Cobza¸s et al., 2019, Theorem 8.4.10], such as the metric induced by k . k H by [Cobza¸s et al., 2019, Theorem 8.5.7]. Hence the restriction of the map-ping H µ ( ν , · ) to this subset is lower semicontinuous, and therefore has pointsof continuity by [Si and Zhang, 2020, Theorem 1.1], so that ∃ ν ∈ cabv ( X ) suchthat H µ ( ν , · ) is continuous at ν .Since the condition [Zalinescu, 2002, Theorem 2.8.1(iii)] is satisfied, by [Zalinescu, 2002,Corollary 2.8.2], one hasinf ν ∈ cabv ( X, { H µ (( ν, ν )) } = max f ∈ Lip ( X ) {− H ∗ µ (( − ( f − f ( x )) , f )) } , (28)or equivalentlyinf ν ∈ cabv ( X, { max {k ν k KR , k µ − ν k T V }} = max f ∈ Lip ( X ) (cid:26) − min λ ∈ [0 , (cid:8) i { f ∈ Lip ( X,x ): k f k L ≤ λ } ( − ( f − f ( x )))+ i { f ∈ Lip ( X ): k f k ∞ ≤ − λ } ( f ) − Z f dµ (cid:27)(cid:27) , (29)where, since k − ( f − f ( x )) k L = k f k L and min x { g ( x ) } = − max x {− g ( x ) } , theright side further simplifies tomax f ∈ Lip ( X ) (cid:26) max λ ∈ [0 , (cid:26)Z f dµ − i { f ∈ Lip ( X,x ): k f k L ≤ λ } ( f ) − i { f ∈ Lip ( X ): k f k ∞ ≤ − λ } ( f ) (cid:27)(cid:27) . (30)5iven f ∈ Lip ( X ), it is clear thatmax λ ∈ [0 , (cid:26)Z f dµ − i { f ∈ Lip ( X,x ): k f k L ≤ λ } ( f ) − i { f ∈ Lip ( X ): k f k ∞ ≤ − λ } ( f ) (cid:27) = (R f dµ if ∃ λ ∈ [0 ,
1] : k f k ∞ ≤ − λ ∧ k f k L ≤ λ , −∞ otherwise. (31)All we need to show now is that ∃ λ ∈ [0 ,
1] : k f k ∞ ≤ − λ ∧ k f k L ≤ λ ⇐⇒ k f k sum ≤ . (32)If k f k ∞ ≤ − λ ∧k f k L ≤ λ , then k f k ∞ + k f k L ≤ λ +(1 − λ ) = 1. If k f k sum ≤ λ = k f k L suffices. Hence one hasinf ν ∈ cabv ( X, { max {k ν k KR , k µ − ν k T V }} = max f ∈ Lip ( X ) , k f k sum ≤ (cid:26)Z f dµ (cid:27) , (33)which is exactly the primal formula we set out to prove, together with thevariant of the dual formula with a maximum instead of a supremum, implyingthat ∃ f ∗ ∈ Lip ( X ) : k f ∗ k sum ≤ ∧k µ k MK = R f ∗ dµ . If k f ∗ k sum < f ∗ , then we get the contradiction R k f ∗ k − sum f ∗ dµ > k µ k MK , so that one actuallyhas k f ∗ k sum = 1.For the density claim, notice that the missing ingredient for the proof of[Cobza¸s et al., 2019, Proposition 8.5.3] to work with k . k MK instead of k . k H isthe formula ∀ ν ∈ cabv ( X, , µ ∈ cabv ( X ) : k µ k MK ≤ k ν k KR + k µ − ν k T V ,which in light of the primal representation clearly holds.For the duality claim, notice again that the missing ingredient for the proof of[Cobza¸s et al., 2019, Proposition 8.5.5] to work with k . k MK instead of k . k H is ex-actly the density claim we just proved, hence ( cabv ( X ) , k . k MK ) ∗ ∼ = ( Lip ( X ) , k . k sum )holds as well.For the equivalence claim, notice that on one hand, one has k ν k KR + k µ − ν k T V ≥ max {k ν k KR , k µ − ν k T V } for ∀ ν ∈ cabv ( X, k µ k H ≥ k µ k MK for ∀ µ ∈ cabv ( X ). On the other hand, one has 2 max {k ν k KR , k µ − ν k T V } ≥k ν k KR + k µ − ν k T V for ∀ ν ∈ cabv ( X, k µ k MK ≥ k µ k H for ∀ µ ∈ cabv ( X ). Therefore the norms k . k MK and k . k H are equivalent. References [Chitescu et al., 2014] Chitescu, I., Ioana, L., Miculescu, R., and Nita, L.(2014). Monge–kantorovich norms on spaces of vector measures.
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