Simple Piecewise Geodesic Interpolation of Simple and Jordan Curves with Applications
SSimple Piecewise Geodesic Interpolation ofSimple and Jordan Curves with Applications
H. Boedihardjo ∗ and X. Geng † November 5, 2018
Abstract
We explicitly construct simple, piecewise minimizing geodesic, arbi-trarily fine interpolation of simple and Jordan curves on a Riemannianmanifold. In particular, a finite sequence of partition points can be spec-ified in advance to be included in our construction. Then we present twoapplications of our main results: the generalized Green’s theorem and theuniqueness of signature for planar Jordan curves with finite p -variationfor (cid:54) p < . Keywords
Piecewise Geodesic Interpolation · Simple Curves · JordanCurves · Generalized Green’s Theorem · Uniqueness of Signature
Mathematics Subject Classification (2000) · · The classical proofs of many properties of Jordan curves (e.g. the Jordan curvetheorem) or functions on Jordan curves (e.g. Cauchy’s theorem) begin with theconsideration of polygonal Jordan curves. As part of the proof of the Jordancurve theorem in [12], it was shown that for every planar Jordan curve, thereis a polygonal Jordan curve that approximates the original Jordan curve arbi-trarily well. Here we shall prove a stronger and more general fact that givena Jordan curve on a connected Riemannian manifold M and n points on thecurve, there exists a simple, piecewise minimizing geodesic, arbitrarily fine in-terpolation which contains these n points as interpolation points. The proofrelies on another main result of this paper for non-closed simple curves. Suchcase was first treated by Werness [13], in which the author used an inductive butnon-constructive method. Here we provide another proof of this result, which ∗ The Oxford-Man Institute, Eagle House, Walton Well Road, Oxford OX2 6ED.Email: [email protected] † Mathematical Institute, University of Oxford, Oxford OX2 6GG and the Oxford-ManInstitute, Eagle House, Walton Well Road, Oxford OX2 6ED.Email: [email protected] a r X i v : . [ m a t h . C A ] J u l as the advantage of being explicit and constructive, and hence numericallycomputable.We would like to emphasize that our approximation, unlike that of [12]which is a direct consequence of our result, does not rely on the flatness of theEuclidean metric, and respects the parametrization of the curve, i.e. it is aninterpolation rather than merely an approximation in the uniform norm. Thelatter is particularly important for applications in the context of rough paththeory, where we approximate continuous paths by bounded variation ones inthe p -variation metric. Such idea is fundamental to the study of the roughnessof continuous paths, and particularly of sample paths of continuous stochasticprocesses (see [4], [6]).We also give two applications of our main result.Taking advantage of the fact that the p -variation of a piecewise linear inter-polation of a path is bounded by the p -variation of the path itself, our approx-imation theorem gives immediately Green’s theorem for planar Jordan curveswith finite p -variation, where (cid:54) p < . To our best knowledge, in the roughpath literature, the only other attempt in extending Green’s theorem to non-rectifiable curves appeared in [14], where Green’s theorem was proved for theboundaries of α -Hölder domains for < α < . Our result is a partial gener-alization of Yam’s. Yam’s result requires the curve to be α -Hölder continuousunder the conformal parametrization whereas our result only requires the curveto be α -Hölder continuous under some parametrization.A long-standing open problem in rough path theory is to what extent a pathcan be determined from its iterated integrals of any order. This is usually knownas the uniqueness of signature problem. Hambly and Lyons [8] proved that theiterated integrals of a rectifiable curve vanish if and only if the path is tree-like,based on a similar type of approximation result for tree-like paths. Using ourapproximation result, we prove the uniqueness of signature for planar Jordancurves with finite p -variations, where (cid:54) p < . The case of non-closed simplecurves was treated in [2]. To our best knowledge this is the strongest uniquenessof signature result so far for non-rectifiable curves.Throughout the rest of this paper, all curves are assumed to be continuous. In this section, we are going to prove our main results about simple piecewisegeodesic approximation of simple and Jordan curves in Riemannian manifolds.Although the most interesting and nontrivial case lies in the Euclidean plane, weformulate our problems in a Riemannian geometric setting of arbitrary dimen-sion since the proofs do not rely on Euclidean geometry (that is, the “flatness”of Euclidean metric) at all.Throughout this section, let M be a d -dimensional connected Riemannianmanifold ( d (cid:62) ). 2he following lemma, which is an easy fact from Riemannian geometry, isfundamental for us to formulate our main results. Lemma 2.1.
For any compact set K ⊂ M , there exists some ε = ε K > , suchthat for any x, y ∈ K with d ( x, y ) < ε, there exists a unique minimizing geodesic in M joining x and y , where d ( · , · ) denotes the Riemannian distance function.Proof. For any x ∈ K, choose δ x small enough such that B ( x, δ x ) is a geodesi-cally convex normal ball (see [5], p. 76, Proposition 4.2). By compactness, wehave a finite covering of K : K ⊂ k (cid:91) i =1 B (cid:18) x i , δ x i (cid:19) , where x , · · · , x k ∈ K. Let ε = min { δ x , · · · , δ x k } . It follows that for any x, y ∈ K with d ( x, y ) < ε, there exists some (cid:54) i (cid:54) k, such that x, y ∈ B ( x i , δ x i ) . Therefore, by geodesic convexity we know that x and y can bejoined by a unique minimizing geodesic in M which lies in B ( x i , δ x i ) . Now we are in position to state our main results.The first main result is a simple piecewise geodesic approximation theoremfor non-closed simple curves in M . Theorem 2.1.
Let γ be a non-closed simple curve in M . For all ε > , thereexists a finite partition P ε : 0 = t < t < · · · < t n − < t n = 1 of [0 , , such that(1) the mesh size of the partition (cid:107)P ε (cid:107) = max i =1 , ··· ,n ( t i − t i − ) < ε ;(2) for any i = 1 , · · · , n, γ t i − and γ t i can be joined by a unique minimizinggeodesic in M , and the piecewise geodesic interpolation (more precisely, piece-wise minimizing geodesic interpolation, and the same thereafter) γ P ε of γ overthe partition points in P ε is a simple curve. The proof of Theorem 2.1 relies on the following crucial lemma, which de-pends heavily on properties of minimizing geodesics. In the Euclidean case, weillustrate the lemma in Figure 1, which says that if the lengths of the straightline segments xy and zw are both less than or equal to r , then at least one ofthe four line segments xz, xw, yz, yw has length strictly less than r . Lemma 2.2.
Let x, y, z, w ∈ M and α : [0 , → M (respectively, β : [0 , → M ) be a minimizing geodesic joining x and y (respectively, z and w ). Assumethat α ([0 , (cid:84) β ([0 , (cid:54) = ∅ and for some r > , d ( x, y ) (cid:54) r, d ( z, w ) (cid:54) r .Then at least one of d ( x, z ) , d ( y, z ) , d ( x, w ) , d ( y, w ) is strictly less than r . xy and zw are lessthan or equal to r . Here the length of zx is strictly less than r . Proof.
Without loss of generality, we shall assume that all geodesics are parametrizedat constant speed. Let α ( u ) = β ( v ) = p for some u, v ∈ [0 , . Since α and β are minimizing geodesics, we know that d ( x, y ) = d ( x, p ) + d ( p, y ) (cid:54) r,d ( z, w ) = d ( z, p ) + d ( p, w ) (cid:54) r. Therefore, at least one of the following four cases happens:(1) d ( x, p ) (cid:54) r , d ( z, p ) (cid:54) r ;(2) d ( x, p ) (cid:54) r , d ( p, w ) (cid:54) r ;(3) d ( p, y ) (cid:54) r , d ( z, p ) (cid:54) r ;(4) d ( p, y ) (cid:54) r ; d ( p, w ) (cid:54) r . First assume that (1) holds. It follows that d ( x, z ) (cid:54) d ( x, p ) + d ( z, p ) (cid:54) r. If d ( x, z ) = r, then d ( x, p ) = d ( z, p ) = r , and hence (4) holds, which implies d ( y, w ) (cid:54) d ( p, y ) + d ( p, w ) (cid:54) r. If d ( y, w ) = r, then d ( p, y ) = d ( p, w ) = r . Consequently, we have u = v = . (cid:101) α ( t ) = (cid:40) α ( t ) , t ∈ (cid:2) , (cid:3) ; β (1 − t ) , t ∈ (cid:2) , (cid:3) . Since Length ( (cid:101) α ) = r = d ( x, z ) , (cid:101) α is minimizing. Moreover, since any geodesic has constant speed, by definitionwe know that (cid:101) α is parametrized proportionally to arc length. It follows fromthe first variation formula (see [5], p. 195, Proposition 2.4) that (cid:101) α must be ageodesic. However, since (cid:101) α | [ , ] = α | [ , ] , by the uniqueness of geodesics wehave (cid:101) α = α and hence y = z. Similarly we have x = w. The other cases can be treated in the same way, which completes the proofof the lemma.With the help of Lemma 2.2, we can now prove Theorem 2.1. The key idea isto construct a sequence of times t , t , . . . , such that t i +1 is the last exit time of γ from a small geodesic ball around γ t i after time t i . The uniform continuity ofthe inverse of the map t → γ t will guarantee that t i and t i +1 are close. We thenneed to argue that adjacent geodesic segments as well as non-adjacent geodesicsegments in the approximation curve do not intersect. The latter uses Lemma2.2. We illustrate the first step of the construction in Figure 2. Proof of Theorem 2.1.
Fix ε > . Since γ is a continuous and injective mappingfrom the compact space [0 , to the Hausdorff space M, it is a homeomorphismfrom [0 , to its image. By compactness and hence uniform continuity of γ − we know that there exists δ ε > such that for any s, t ∈ [0 , ,d ( γ s , γ t ) < δ ε = ⇒ | t − s | < ε. We further assume that δ ε < ε γ ([0 , , where ε γ ([0 , is the positive number inLemma 2.1 depending on the compact set γ ([0 , ⊂ M. It follows from Lemma2.1 that for any s, t ∈ [0 , with d ( γ s , γ t ) < δ ε , γ s and γ t can be joined by aunique minimizing geodesic in M . Now define an increasing sequence of points { t i } ∞ i =0 in [0 , inductively by setting t = 0 and t i = sup (cid:26) t ∈ [ t i − ,
1] : γ t ∈ B (cid:18) γ t i − , δ ε (cid:19)(cid:27) , i (cid:62) . We claim that there exists some l (cid:62) , such that for all i (cid:62) l , t i = 1 . In fact,if it is not the case, then for any i (cid:62) , we have t i − < t i < and d (cid:0) γ t i − , γ t i (cid:1) = δ ε . On the other hand, by the uniform continuity of γ, there exists some η ε > , such that for any s, t ∈ [0 , , | t − s | < η ε = ⇒ d ( γ s , γ t ) < δ ε . γ . The solidgeodesic segment joining the point γ and γ t represents the first step in theconstruction of the piecewise geodesic interpolation of γ . Note that we take t to be the last exit time of γ from a δ ε -geodesic ball centered at γ .Therefore, for any i (cid:62) , | t i − t i − | (cid:62) η ε , which is an obvious contradiction.Now set l = min { i (cid:62) t i = 1 } , and define P ε : 0 = t < t < · · · < t l − < t l = 1 to be a finite partition of [0 , . Then it is easy to see that (cid:107)P ε (cid:107) < ε , where (cid:107)P ε (cid:107) denote the mesh size of the partition P ε .It remains to show that the piecewise geodesic interpolation γ P ε of γ overthe points of P ε is a simple curve.To see this, first notice that for adjacent intervals [ t i − , t i ] , [ t i , t i +1 ] , wehave γ P ε | [ t i − ,t i ] (cid:92) γ P ε | [ t i ,t i +1 ] = { γ t i } . In fact, if it is not the case, then there exist s ∈ [ t i − , t i ) and s ∈ ( t i , t i +1 ] such that γ P ε s = γ P ε s (cid:54) = γ t i . If i < l − , then by applying Lemma 2.1 with x = γ t i and y = γ P ε s = γ P ε s , γ P ε | [ s ,t i ] is a reparametrization of the reversal of γ P ε | [ t i ,s ] , which we denote as ←−− γ P ε | [ t i ,s ] . In particular, γ P ε | [ t i ,t i +1 ] and ←−− γ P ε | [ t i − ,t i ] are geodesics that starts atthe same position with the same initial velocity. By the uniqueness of geodesic,either γ P ε ([ t i , t i +1 ]) ⊆ ←−− γ P ε ([ t i − , t i ]) or ←−− γ P ε ([ t i − , t i ]) ⊆ γ P ε ([ t i , t i +1 ]) . Inparticular we have either γ P ε | [ t i − ,t i ] passes through γ t i +1 or γ P ε | [ t i ,t i +1 ] passesthrough γ t i − . As ←−− γ P ε | [ t i − ,t i ] and γ P ε | [ t i ,t i +1 ] are minimizing geodesics andwe have d (cid:0) γ t i , γ t i − (cid:1) = d (cid:0) γ t i , γ t i +1 (cid:1) , we conclude that γ t i − = γ t i +1 whichcontradicts that γ is simple. Figure 3 illustrates this argument.6igure 3: This figure illustrates the argument in the proof of Theorem 2.1 thattwo adjacent line segment of the approximation curve we constructed cannotintersect. The straight line represents the geodesic segments in the piecewisegeodesic interpolation of γ . γ t i − , γ t i , γ t i +1 are subdivision points of the curve.If the two adjacent line segments do intersect, as in the figure below, then γ t i +1 would be closer to γ t i − than to γ t i which would contradict our construction.If i = l − , then arguing as in the case i < l − , we have either γ P ε ([ t i , t i +1 ]) ⊆←−− γ P ε ([ t i − , t i ]) or ←−− γ P ε ([ t i − , t i ]) ⊆ γ P ε ([ t i , t i +1 ]) . However, as i = l − , we have d (cid:0) γ t i , γ t i +1 (cid:1) (cid:54) δ ε = d (cid:0) γ t i − , γ t i (cid:1) and hence γ P ε ([ t i , t i +1 ]) ⊆ ←−− γ P ε ([ t i − , t i ]) .In particular, γ P ε | [ t i − ,t i ] passes through γ t i +1 . Therefore, d (cid:0) γ t i − , γ t i +1 (cid:1) (cid:54) d (cid:0) γ t i − , γ t i (cid:1) which contradicts the construction of { t i } li =0 . On the other hand, if [ t i − , t i ] and [ t j − , t j ] ( i < j ) are non-adjacent intervalsand γ P ε | [ t i − ,t i ] (cid:92) γ P ε | [ t j − ,t j ] (cid:54) = ∅ , then by Lemma 2.2 we know that at least one of d (cid:0) γ t i − , γ t j − (cid:1) , d (cid:0) γ t i , γ t j − (cid:1) , d (cid:0) γ t i − , γ t j (cid:1) , d (cid:0) γ t i , γ t j (cid:1) is strictly less than δ ε . However, this again contradicts the construction of { t i } li =0 . Now the proof is complete.The same technique of proof will allow us to prove our second main result,which is concerned with simple piecewise geodesic approximations of Jordancurves. This result significantly strengthens Theorem 2.1.
Theorem 2.2.
Let γ : [0 , → M be a Jordan curve. Assume that < τ < · · · < τ k < are k fixed points in [0 , . Then for any ε > , there exists a finitepartition P ε : 0 = t < t < · · · < t n − < t n = 1 of [0 , , such that (1) τ , · · · , τ k are partition points of P ε ;(2) (cid:107)P ε (cid:107) < ε ;(3) for i = 1 , · · · , n, γ t i − and γ t i can be joined by a unique minimizinggeodesic in M , and the piecewise geodesic interpolation γ P ε of γ over the parti-tion points in P ε is a Jordan curve. The proof of Theorem 2.2 relies on the following geometric fact. It is illus-trated by Figure 4.
Lemma 2.3.
Let B ( p, R ) be a geodesically convex normal ball centered at p ∈ M , and let q ∈ ∂B ( p, R ) . Assume that x, y ∈ B ( p, R ) c and there exists aminimizing geodesic α : [0 , → M joining x and y. If α ([0 , (cid:84) pq (cid:54) = ∅ and d ( x, y ) (cid:54) r for some < r < R, where pq denotes the image of the uniqueminimizing geodesic in M joining p and q , then d ( x, q ) < r, d ( y, q ) < r. Proof.
The conclusion is obvious if q ∈ α ([0 , . Otherwise, let t ∈ (0 , be theunique time such that e := α ( t ) ∈ B ( p, R ) is the intersection point of α ([0 , and pq . By using the fact that B ( p, R ) is a geodesically convex normal ball,it is easy to show that there exists a unique u ∈ (0 , t ) and a unique v ∈ ( t, , such that z := α ( u ) and w := α ( v ) lie on ∂B ( p, R ) . Observe that e and q aredistinct, for their equality would contradict the fact that r < R. Now it follows8rom properties of minimizing geodesics that d ( x, q ) (cid:54) d ( x, e ) + d ( e, q )= d ( x, e ) + d ( p, q ) − d ( p, e )= d ( x, e ) + d ( p, w ) − d ( p, e ) (cid:54) d ( x, e ) + d ( e, w )= d ( x, w ) < d ( x, y ) (cid:54) r. Similarly, we have d ( y, q ) < r. Now we can prove Theorem 2.2. Our proof is constructive and the idea isas follows. Recall that the times τ , . . . , τ k should to included in our partition.Firstly, We find small disjoint geodesic balls around the points γ τ i , . . . , γ τ k , γ .Secondly, we connect each point γ τ i by two radial minimizing geodesics to thepoint where γ first enters the geodesic ball around γ τ i before time τ i and tothe point where γ last exists the geodesic ball. Finally, we construct simplepiecewise geodesic interpolation for each piece of simple curves outside thosegeodesic balls inductively, by using the algorithm in Theorem 2.1. To makesure that those approximation curves do not intersect the geodesic segmentsinside those geodesic balls, we need to use Lemma 2.3. Figure 5 illustrates theidea when k = 2 . Proof of Theorem 2.2.
Take an arbitrary τ ∈ (0 , τ ) . Since γ is a Jordan curve,we know that γ τ , γ τ , · · · , γ τ k , γ τ k +1 ∈ M are all distinct, where we set τ k +1 = 1 .By the Hausdorff property, there exists some δ > such that the closed metricballs B ( γ τ , δ ) , · · · , B (cid:0) γ τ k +1 , δ (cid:1) are all disjoint and γ τ / ∈ (cid:83) k +1 i =1 B ( γ τ i , δ ) .For the moment, by periodic extension and restriction we regard γ as definedon [ τ, τ + 1] with starting and end points being γ ( τ ) . Now fix ε > . Without loss of generality we assume that ε < min (cid:26) τ, τ − τ, τ − τ , · · · , τ k +1 − τ k (cid:27) . First of all, by the uniform continuity of γ | − τ,τ i ] and γ | − τ i ,τ +1] , there existssome δ ε > , such that for all i = 1 , · · · , k +1 , any s, t ∈ [ τ, τ i ] or s, t ∈ [ τ i , τ + 1] ,d ( γ s , γ t ) < δ ε = ⇒ | t − s | < ε. Now set U i = B ( γ τ i , δ ε ) . Here we assume that δ ε is small enough so that each U i is a geodesically convex normal ball and Lemma 2.1 holds for those γ s , γ t with d ( γ s , γ t ) < δ ε . Define u i = inf (cid:8) t ∈ [ τ, τ i ] : γ t ∈ U i (cid:9) ,v i = sup (cid:8) t ∈ [ τ i , τ + 1] : γ t ∈ U i (cid:9) . k = 2 . The dotted line represents the curve γ . The solid line represents the piecewisegeodesic interpolation of γ .To return to the original time interval [0 , , let v = v k +1 − . We have | v | < ε , | τ i − u i | < ε , | v i − τ i | < ε and < v < u < τ < v < · · · < u k < τ k < v k < u k +1 < , and γ u i (cid:54) = γ v i , d ( γ τ i , γ u i ) = d ( γ τ i , γ v i ) = δ ε . Moreover, we have γ | ( v ,u ) ∪ ( v ,u ) ∪···∪ ( v k − ,u k ) ∪ ( v k ,u k +1 ) (cid:92) (cid:32) k +1 (cid:91) i =1 U i (cid:33) = ∅ . We will take v , u , τ , v , · · · , u k , τ k , v k , u k +1 as part of the partition pointsin P ε . In particular, v will be the first point, u k +1 will be the last point(except and ), and u i , τ i , v i will be successive points in P ε , so the piecewisegeodesic interpolation of γ over those small intervals is a finite sequence of radialgeodesics of the balls centered at γ τ i with radius δ ε for i = 1 , · · · , k + 1 .For the next step, notice that γ | [ v ,u ] , γ | [ v ,u ] , · · · , γ | [ v k ,u k +1 ] are k + 1 non-closed simple curves with disjoint images. We are going to use the constructiveprocedure in the proof of Theorem 2.1 to define a simple piecewise geodesicapproximation of each γ | [ v i − ,u i ] ( i = 1 , · · · , k + 1 ) with partition size smallerthan ε inductively, such that the resulting piecewise geodesic closed curve over [0 , is simple. That will complete the proof of the theorem.Let γ (0) be the Jordan curve such that γ (0) = γ, on [ v , u ] ∪ [ v , u ] ∪ · · · ∪ [ v k − , u k ] ∪ [ v k , u k +1 ] , [0 , v ] , [ u , τ ] , [ τ , v ] , · · · , [ u k , τ k ] , [ τ k , v k ] , [ u k +1 , . By the construction in the proof of Theorem 2.1, we may find a partition P (1)[ v ,u ] : v = w (1)0 < w (1)1 < · · · < w (1) l − < w (1) l = u so that (cid:13)(cid:13)(cid:13) P (1)[ v ,u ] (cid:13)(cid:13)(cid:13) < ε , the geodesic interpolation γ P (1)[ v ,u of γ | [ v ,u ] over thepartition points in P (1)[ v ,u ] is simple and d (cid:16) γ w (1) i − , γ w (1) i (cid:17) = δ (1) ε , i = 1 , · · · l − ,d (cid:18) γ w (1) l − , γ u (cid:19) (cid:54) δ (1) ε , for some δ (1) ε > .Moreover, we may choose δ (1) ε small enough so that dist (cid:18) γ P (1)[ v u , γ (0) | [ τ , (cid:19) > and δ (1) ε < δ ε .Now we will show that γ P (1)[ v ,u (cid:92) γ (0) | [0 ,v ) ∪ ( u ,τ ] = ∅ . In fact, if γ P (1)[ v ,u (cid:84) γ (0) | [0 ,v ) (cid:54) = ∅ , then from the construction of (cid:110) w (1) i (cid:111) , thereexists some i (cid:62) , such that γ w (1) i − , γ w (1) i ∈ U k +1 c and γ w (1) i − γ w (1) i (cid:92) γ (0) | [0 ,v ) (cid:54) = ∅ , where γ w (1) i − γ w (1) i denotes the image of the unique minimizing geodesic joining γ w (1) i − and γ w (1) i . However, since d (cid:16) γ w (1) i − , γ w (1) i (cid:17) (cid:54) δ (1) ε < δ ε , we know fromLemma 2.3 that d (cid:16) γ v , γ w (1) i − (cid:17) < δ (1) ε , d (cid:16) γ v , γ w (1) i (cid:17) < δ (1) ε , which is an obvious contradiction to the construction of (cid:110) w (1) i (cid:111) l i =0 . On theother hand, if γ P (1)[ v ,u (cid:84) γ (0) | ( u ,τ ] (cid:54) = ∅ , then there exists some i (cid:54) l − , suchthat γ w (1) i − γ w (1) i (cid:84) γ (0) | ( u ,τ ] (cid:54) = ∅ . Since γ w (1) i − , γ w (1) i ∈ U c and d (cid:16) γ w (1) i − , γ w (1) i (cid:17) = δ (1) ε < δ ε , we know againfrom Lemma 2.3 that d (cid:16) γ u , γ w (1) i − (cid:17) < δ (1) ε , d (cid:16) γ u , γ w (1) i (cid:17) < δ (1) ε . (cid:110) w (1) i (cid:111) l i =0 . Therefore, the closed curve γ (1) over [0 , defined by γ (1) t = γ P (1)[ v ,u t , t ∈ [ v , u ] ; γ (0) t , t ∈ [0 , \ [ v , u ] , is a Jordan curve.Now consider γ | [ v ,u ] . The previous argument can be carried through easilywith respect to the Jordan curve γ (1) , and we obtain a finite partition P (2)[ v ,u ] : v = w (2)0 < w (2)1 < · · · < w (2) l − < w (2) l = u , such that (cid:13)(cid:13)(cid:13) P (2)[ v ,u ] (cid:13)(cid:13)(cid:13) < ε, and the closed curve γ (2) over [0 , defined by γ (2) t = γ P (2)[ v ,u t , t ∈ [ v , u ] ; γ (1) t , t ∈ [0 , \ [ v , u ] , is a Jordan curve, where γ P (2)[ v ,u is the geodesic interpolation of γ | [ v ,u ] overthe partition points in P (2)[ v ,u ] . By induction, we are able to construct simplepiecewise geodesic approximation of each piece of γ outside ∪ k +1 i =1 U i and finallyobtain a finite partition P ε of [0 , with partition points { } (cid:91) (cid:32) k +1 (cid:91) i =1 (cid:110) v i − , w ( i )1 , · · · , w ( i ) l i − , u i , τ i (cid:111) , (cid:33) such that (cid:107)P ε (cid:107) < ε, and the geodesic interpolation γ P ε (which is γ ( k +1) byinduction) of γ over the points of P ε is a Jordan curve.Now the proof is complete. Remark . By slight modification of the proof, it is not hard to see thatTheorem 2.2 also holds for non-closed simple curves. In this case, it strengthensthe result of Theorem 2.1.
Remark . It is possible to generalize our main results to infinite dimensionalspaces with suitable geodesic properties. For technical simplicity we are notgoing to present the details.
In this section, we shall demonstrate two applications of Theorem 2.2. Here weassume that M = R . .1 Green’s theorem for Jordan curves with finite p -variation( (cid:54) p < ) We will prove a generalized Green’s theorem for planar Jordan curves withfinite p -variation, where (cid:54) p < . First we shall briefly recall basic facts aboutYoung’s integration. Definition 3.1.
Let ( X, d ) be a metric space. We say a function γ : [0 , → X has finite p -variation if (cid:107) γ (cid:107) p := (cid:32) sup P (cid:88) t Let γ : [0 , → R d be a path with finite p -variation, where p (cid:62) .Let f : R d → R ˜ d be a Lipschitz function with Lipschitz constant C . Then1. ([11], Lemma 1.12 and Lemma 1.18) (cid:107) f ( γ ) (cid:107) p (cid:54) C (cid:107) γ (cid:107) p .2. ([11], Proposition 1.14 and Remark 1.19) For all q > p , (cid:13)(cid:13) f ( γ ) − f (cid:0) γ P (cid:1)(cid:13)(cid:13) q → as (cid:107)P(cid:107) → . We now prove Green’s theorem for non-smooth Jordan curves. Theorem 3.2. Let f, g : R → R be functions with continuous first orderderivatives, and let γ : [0 , → R be a positively oriented Jordan curve withfinite p -variation, where (cid:54) p < . Let x · , y · denote the first and secondcoordinate components of γ · respectively. Then ˆ ( f ( γ s ) d y s − g ( γ s ) d x s ) = ˆ Int( γ ) (cid:18) ∂f∂x + ∂g∂y (cid:19) d x d y, here the integral on the L.H.S. is understood as the Young’s integral, and Int ( γ ) denotes the interior of γ. Proof. Fix ε > . According to Theorem 2.2, let P ε be a finite partition of [0 , such that (cid:107)P ε (cid:107) < ε, and the piecewise linear interpolation γ P ε of γ over thepartition points in P ε is a Jordan curve. Let x P ε , y P ε be the first and secondcomponents of γ P ε respectively. It follows from the classical Green’s theoremfor piecewise smooth Jordan curve that ˆ ( f (cid:0) γ P ε s (cid:1) d y P ε s − g (cid:0) γ P ε s (cid:1) d x P ε s ) = ˆ Int( γ P ε ) (cid:18) ∂f∂x + ∂g∂y (cid:19) d x d y. For any q ∈ ( p, , we know that (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( f (cid:0) γ P ε s (cid:1) d y P ε s − ˆ f ( γ s ) d y s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ (cid:0) f (cid:0) γ P ε s (cid:1) − f ( γ s ) (cid:1) d y P ε s + ˆ f ( γ s ) d (cid:0) y P ε s − y s (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12)(cid:12) ˆ (cid:0) f (cid:0) γ P ε s (cid:1) − f ( γ s ) (cid:1) d y P ε s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˆ f ( γ s ) d (cid:0) y P ε s − y s (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) ζ (cid:18) q (cid:19) (cid:16)(cid:13)(cid:13) f (cid:0) γ P ε · (cid:1) − f ( γ · ) (cid:13)(cid:13) q (cid:107) γ (cid:107) q + (cid:107) f ( γ · ) (cid:107) q (cid:13)(cid:13) γ P ε · − γ · (cid:13)(cid:13) q (cid:17) , where the final inequality follows from Theorem 3.1 and Lemma 3.1. Therefore,by Lemma 3.1, ˆ f (cid:0) γ P ε s (cid:1) d y P ε s → ˆ f ( γ s ) d y s as ε → . Similarly, ˆ g (cid:0) γ P ε s (cid:1) d y P ε s → ˆ g ( γ s ) d y s as ε → .On the other hand, as γ has finite p variation, it has a p -Hölder parametriza-tion. Therefore, γ has Hausdorff dimension less than . In particular, this meansthat γ ([0 , has zero Lebesgue measure. By applying the bounded convergencetheorem to the integrand (cid:18) ∂f∂x + ∂g∂y (cid:19) Int( γ P ε ) , we have ˆ Int( γ P ε ) (cid:18) ∂f∂x + ∂g∂y (cid:19) d x d y → ˆ Int( γ ) (cid:18) ∂f∂x + ∂g∂y (cid:19) d x d y as ε → .Now the proof is complete. 14 emark . For rectifiable paths, there is a version of Green’s theorem whichworks for non-simple closed curves, involving the winding number of a path.An interesting inequality in this respect is the Banchoff-Pohl inequality (see[1]), which generalizes the isoperimetric inequality and asserts that the windingnumber of a rectifiable curve is square-integrable. The reason for the “simpleclosed” condition in our version of Green’s theorem is because in general thewinding number of a non-simple non-rectifiable curve is not integrable. Thefact that we can approximate the rough Jordan curves by piecewise linear inter-polations which are still Jordan means that the winding number of each approx-imation is an indicator function, which is bounded by the indicator function ofa neighborhood of Int ( γ ) . Remark . A direct consequence of Theorem 3.2 is Cauchy’s theorem forJordan curves with finite p -variation where (cid:54) p < , according to the Cauchy-Riemann equation for holomorphic functions. p -variation ( (cid:54) p < ) The sequence of iterated integrals (formally known as the signature) of pathsplays a key role in rough path theory. A central open problem in this area is todetermine a path from its iterated integrals. In the case of bounded variationpaths, Hambly and Lyons [8] proved that two paths of bounded variation canhave the same sequence of iterated integrals if and only if they can be obtainedfrom each other by a ”tree-like deformation”. In [2], it was proved that for planarsimple curves with finite p -variation, the sequence of iterated integrals of thepath determines the path up to reparametrization. In the context of stochasticprocesses, it was proved in [9] that the sequence of iterated Stratonovich’s inte-grals of Brownian motion determines the Brownian sample paths almost surely.This result was extended to diffusion processes in [7].Here we are going to prove the uniqueness of signature for planar Jordancurves with finite p -variation, where (cid:54) p < is fixed throughout the rest ofthis section.We shall follow [10] and embed the sequence of iterated integrals of a pathinto the tensor algebra, which gives us a very nice algebraic structure to workwith.Assume that (cid:0) R d (cid:1) ⊗ n is the tensor produce space equipped with the Eu-clidean norm by identifying it with R d n . Let T (cid:0) R d (cid:1) = ⊕ ∞ n =0 (cid:0) R d (cid:1) ⊗ n , and let π n : T (cid:0) R d (cid:1) → (cid:0) R d (cid:1) ⊗ n denote the projection map. Assume that { e , . . . , e d } is the standard basis of R d , and { e ∗ , . . . , e ∗ d } is the correspond-ing dual basis of R d ∗ . We embed T (cid:16)(cid:0) R d (cid:1) ∗ (cid:17) into T (cid:0) R d (cid:1) ∗ by extending the15elation e ∗ i ⊗ . . . ⊗ e ∗ i n ( e j ⊗ . . . ⊗ e j k ) = (cid:40) if n = k and i = j , · · · , i k = j k , otherwise,linearly. Definition 3.2. Let γ : [0 , → R d be a continuous path with finite p -variation,then the formal series of tensors S ( γ ) , := 1 + ∞ (cid:88) i =1 ˆ Let γ : [0 , → R d be a continuous path with finite p -variation. Then1. ([11], p. 32) Let r : [0 , → [0 , be a continuous increasing function,then S ( γ · ) , = S (cid:0) γ r ( · ) (cid:1) , . 2. ([11], p. 32) For all a ∈ R d , S ( a + γ · ) , = S ( γ · ) , . 3. ([11], Corollary 2.11) Let γ n be a sequence of paths with finite p -variationand (cid:107) γ − γ n (cid:107) p → , as n → ∞ , then for each k ∈ N , (cid:12)(cid:12)(cid:12) π k (cid:16) S ( γ n ) , (cid:17) − π k ( S ( γ )) (cid:12)(cid:12)(cid:12) → as n → ∞ . It turns out that some terms in the signature of a curve can be reduced toa single line integral. This is the key idea to prove our uniqueness of signatureresult. Proposition 3.2. Let γ be a positively oriented Jordan curve with finite p -variation. Let x · , y · be the first and second coordinate components of γ respec-tively. Then for any k, n (cid:62) e ∗⊗ ( k +1)1 ⊗ e ∗⊗ ( n +1)2 (cid:16) S ( γ ) , (cid:17) = ˆ ˆ s n + k +2 . . . ˆ s d x s . . . d x s k +1 d y s k +2 . . . d y s n + k +2 = 1 k ! n ! ˆ Int( γ ) ( x − x ) k ( y − y ) n d x d y. roof. Note that ˆ ˆ s n + k +2 . . . ˆ s d x P ε s . . . d x P ε s k +1 d y P ε s k +2 . . . d y P ε s n + k +2 = 1( k + 1)! n ! ˆ (cid:16) x P ε s k +1 − x P ε (cid:17) k +1 (cid:16) y P ε − y P ε s k +1 (cid:17) n d y P ε s k +1 = 1 k ! n ! ˆ Int( γ P ε ) ( x − x ) k ( y − y ) n d x d y. By Lemma 3.1, ˆ ˆ s n + k +2 . . . ˆ s d x P ε s . . . d x P ε s k +1 d y P ε s k +2 . . . d y P ε s n + k +2 → ˆ ˆ s n + k +2 . . . ˆ s d x s . . . d x s k +1 d y s k +2 . . . d y s n + k +2 as ε → .As in the proof of Theorem 3.2, ˆ Int( γ P ε ) ( x − x ) k ( y − y ) n d x d y → ˆ Int( γ ) ( x − x ) k ( y − y ) n d x d y as ε → . Therefore, the result follows. Remark . The case of k = 1 , n = 0 for Proposition 3.2 has been provedby Werness [13]. The main difficulty in extending to the general case involvesmainly the interchange of iterated integrals.The following lemma is the main reason why our result only works for Jordancurves. Lemma 3.2. Let γ, (cid:101) γ : [0 , → R be two positively oriented Jordan curvessuch that γ ([0 , (cid:101) γ ([0 , and γ = (cid:101) γ . There exists a continuous increasingfunction r : [0 , → [0 , such that γ r ( t ) = (cid:101) γ . In other words, γ and (cid:101) γ are equalup to a reparametrization.Proof. As γ is a Jordan curve, γ ([0 , \ γ and (cid:101) γ ([0 , \ (cid:101) γ are both homeomor-phic to (0 , . Therefore, the function r : (0 , → (0 , defined by r ( t ) = γ − ◦ (cid:101) γ ( t ) is a homeomorphism (0 , → (0 , . Hence, it is strictly mono-tone. This implies that lim t → r ( t ) exists. Moreover, it is easy to see that lim t → r ( t ) ∈ { , } .If lim t → r ( t ) = 0 , then r can be extended to a continuous increasing functionon [0 , . As γ r ( t ) = (cid:101) γ t , we know that γ and (cid:101) γ equal up to reparametrization. If lim t → r ( t ) = 1 , then lim t → r ( t ) = 0 and r ( t ) is decreasing. This implies that r (1 − t ) is an increasing continuous function. Therefore, γ and (cid:101) γ have oppositeorientations, which is a contradiction.Now we are in position to state and prove our result on the uniqueness ofsignature for planar Jordan curves. 17 heorem 3.3. Let γ, ˜ γ : [0 , → R be a Jordan curves with finite p -variation.Then S ( γ ) , = S (˜ γ ) , if and only if γ and ˜ γ is a translation and a reparametriza-tion of each other.Proof. Sufficiency follows from Proposition 3.1. We now consider the necessitypart.By applying a translation we may assume that γ = γ = ˜ γ = ˜ γ = 0 .As e ∗ ⊗ e ∗ (cid:16) S ( γ ) , (cid:17) = e ∗ ⊗ e ∗ (cid:16) S (˜ γ ) , (cid:17) , by Proposition 3.2 we have ( − ε ( γ ) ˆ Int( γ ) d x d y = ( − ε (˜ γ ) ˆ Int(˜ γ ) d x d y, where ε ( γ ) is if γ is positively oriented and otherwise. As ´ Int( γ ) d x d y and ´ Int(˜ γ ) d x d y are both positive, we must have γ and ˜ γ oriented in the samedirection.Without loss of generality, assume both γ and ˜ γ are positively oriented. ByProposition 3.2 and that S ( γ ) , = S (˜ γ ) , , we have ˆ Int( γ ) ( x − x ) k ( y − y ) n d x d y = ˆ Int(˜ γ ) ( x − ˜ x ) k (˜ y − y ) n d x d y for all k, n ≥ . Therefore, ˆ Int( γ ) e i ( λ x + λ y ) d x d y = ˆ Int(˜ γ ) e i ( λ x + λ y ) d x d y for all λ , λ ∈ R .Both Int( γ ) and Int(˜ γ ) are in L and by the injectivity of the Fourier trans-form on L , we have Int( γ ) ( x, y ) = Int(˜ γ ) ( x, y ) for almost every ( x, y ) ∈ R . In particular, this implies that both Int ( γ ) \ Int (˜ γ ) ⊂ Int ( γ ) \ Int (˜ γ ) and Int (˜ γ ) \ Int ( γ ) ⊂ Int ( γ ) \ Int (˜ γ ) are null sets in R . How-ever, since both Int (˜ γ ) \ Int ( γ ) and Int ( γ ) \ Int (˜ γ ) are open, they must beempty. Therefore, Int (˜ γ ) = Int ( γ ) . By the Jordan curve theorem, we have R \ Int (˜ γ ) = R \ Int (˜ γ ) . Therefore, Int (˜ γ ) = Int ( γ ) and so γ ([0 , γ ([0 , and by Lemma 3.2, γ and ˜ γ are equal up to reparametrization. Remark . The proof of Theorem 3.3 gives a very explicit way of computingthe moments of the finite measure Int( γ ) ( x, y ) d x d y from the signature of γ. However, due to possible difficulty of numerically inverting the Fourier trans-form, this is still not an explicit reconstruction scheme for a Jordan curve fromits signature. In fact, an explicit reconstruction scheme in general remains asignificant open problem in rough path theory.18 cknowledgement The authors wish to thank Professor Terry Lyons for his valuable suggestionson the present paper. The authors are supported by the Oxford-Man Instituteat University of Oxford. The first author is also supported by ERC (GrantAgreement No.291244 Esig). References [1] T.F. Banchoff and W.F. Pohl, A generalization of the isoperimetric inequal-ity, J. Differential Geom., 6 (2), pp. 175-192.[2] H. Boedihardjo, H. Ni and Z. Qian, Uniqueness of signature for simplecurves, arXiv:1304.0755.[3] S.S. Chern, W. Chen and K.S. Lam, Lectures on differential geometry ,World Scientific, 1999.[4] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and frac-tional Brownian motions, Probab. Theory Related Fields, 122 (2002), pp.108-140.[5] M.P. Do Carmo, Riemannian geometry , Springer, 1992.[6] P.K. Friz and N.B. Victoir, Multidimensional stochastic processes as roughpaths , Cambridge University Press, 2010.[7] X. Geng and Z. Qian, On the Uniqueness of Stratonovich’s Signatures ofMultidimensional Diffusion Paths, arXiv:1304.6985.[8] B. Hambly and T. Lyons, Uniqueness for the signature of a path of boundedvariation and the reduced path group, Ann. of Math., 171 (2010), pp. 109-167.[9] Y. Le Jan and Z. Qian, Stratonovich’s signatures of Brownian motion deter-mine Brownian sample paths, Probab. Theory Related Fields, 157 (2012),pp. 209-223.[10] T. Lyons, Differential equations driven by rough signals, Rev. Mat.Iberoamericana, 14 (1998), pp. 215-310.[11] T. Lyons, M. Caruana, and T. Levy. Differential equations driven by roughpaths , Lecture Note in Mathematics, Vol. 1908, Springer, Berlin, 2007.[12] H. Tverberg, A proof of the Jordan curve theorem, Bull. Lond. Math. Soc.,12 (1980), pp. 34-38.[13] B. Werness, Regularity of Schramm-Loewner evolutions, annular crossings,and rough path theory, Electron. J. Probab., 17 (2012), pp. 1-21.1914] P. Yam,