Characterization of Low Dimensional $RCD^*(K,N)$ spaces

In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RCD^*(K,N)$ spaces) with \emph{non-empty} one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with $Ric \ge K$ and Hausdorff dimension $N$ and the class of $RCD^*(K,N)$ spaces coincide for $N<2$ (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality ( that is ,roughly speaking, a converse to the L\'{e}vy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.


INTRODUCTION
In the past few decades, understanding Ricci limit spaces has been a central theme in geometric analysis. Ricci limit spaces are the metric spaces that are obtained as the pointed Gromov-Hausdorff limits of sequences of Riemannian manifolds with uniform lower Ricci curvature bounds. Studying Ricci limit spaces is a key in understanding the metric and measure properties of Riemannian manifold with lower Ricci curvature bound. A deep theory of these spaces has been developed over the years mostly by the work of Cheeger and Colding (see [11][12][13][14]). and equal to R (for a precise definition of R 1 , see Definition 3.1). We use the structure theory developed by Mondino-Naber [32] and arguments similar to Honda [26] to prove the following characterization theorem. Theorem 1.1. Let (X, d, m) be an RCD * (K, N) space for K ∈ R and N ∈ (1, ∞). Assume X is not one point and supp m = X. The following are all equivalent to each other: (1) m(R j ) = 0 for any j ≥ 2, (4) X is isometric to R, to R ≥0 , to S 1 (r) := {x ∈ R 2 ; |x| = r} for r > 0, or to [0, l] for l > 0.
Moreover the measure m is equivalent to the 1-dimensional Hausdorff measure H 1 i.e. m can be written in the form m = e − f H 1 for a (K, N)-convex function f (see Definition 2.1).
In particular dim H X ∈ Z ≥0 if (X, d, m) is an RCD * (K, N) space that has R 1 = ∅.
A direct corollary is the following: Corollary 1.2. Let (X, d, m) be an RCD * (K, N) space for K ∈ R and N ∈ [1,2). Then the statements as in Theorem 1.1 hold. Remark 1.3. On Ricci limit spaces, the conditions in Theorem 1.1 are also equivalent to 1 ≤ dim H X < 2 ( [17,27] ). So far we do not know whether an RCD * (K, N) space of the Hausdorff dimension n < N has the regular set R k , n < k ≤ N or not.
In order to further understand the behaviour of the measure, we first show the following important Bishop-Gromov type inequality for RCD * (K, N) spaces that was previously known for Ricci limit spaces [26]. Let W E 1 be the set of points where there exists a tangent space of the form R × W for some proper space, W of strictly positive diameter (for a precise definition of W E 1 , see Definition 3.1). Using the Bishop-Gromov type inequality (Theorem 1.4), we will prove the following. and furthermore, if the modulus of continuity of x → m(B r (x)) r is independent of the choice of r ≥ 0 then, M 1 is closed. Remark 1. 6. It has been brought to our attention that recently, a similar result for Ricci limit spaces has been proven in Chen [15]. The proof in Chen [15] heavily relies on the Hölder continuity of tangent cones along a minimal geodesic which is a result that is not available in our setting (RCD * (K, N) metric measure spaces).

PRELIMINARIES
A metric measure space is a triple (X, d, m) consisting of a complete separable metric space, (X, d), and a locally finite complete positive Borel measure, m, that is, m(B) < ∞ for any bounded Borel set B and supp m = ∅.
A curve γ : [0, l] → X is called a geodesic if d(γ(0), γ(l)) = Length(γ). We call (X, d) a geodesic space if for any two points, there exists a geodesic connecting them. A metric space (X, d) is said to be proper if every bounded closed set in X is compact. It is well-known that complete locally compact geodesic metric spaces are proper.
We denote the set of all Lipschitz functions in X by LIP(X). For every f ∈ LIP(X), the local Lipschitz constant at x, |D f |(x), is defined by The Cheeger energy of a function f ∈ L 2 (X, m) is defined as Set D(Ch) := { f ∈ L 2 (X, m) ; Ch( f ) < ∞}. It is known that for any f ∈ D(Ch), there exists |D f | w ∈ L 2 (X, m) such that 2Ch( f ) = X |D f | 2 w dm. We say that (X, d, m) is infinitesimally Hilbertian if the Cheeger energy is a quadratic form. The infinitesimal Hilbertianity is equivalent to the Sobolev space W 1,2 (X, d, m) := { f ∈ L 2 ∩ D(Ch)} equipped with the norm f 2 1,2 := f 2 2 + 2Ch( f ) being a Hilbert space.
2.1. The curvature-dimension conditions. Let (X, d, m) be a metric measure space and P (X), the set of all Borel probability measures. We denote by P 2 (X), the set of all Borel probability measures with finite second moments.
Since we will use the definition of the dimension-less curvature-dimension conditions (namely, CD(K, ∞) conditions) in a few places in this paper, we will recall it here: 1] connecting them such that holds for any t ∈ [0, 1]. We say that (X, d, m) satisfies the strong CD(K, ∞) condition if (2.3) holds for any geodesic. Moreover (X, d, m) is called an RCD(K, ∞) space if it is infinitesimally Hilbertian and a CD(K, ∞) space.
An important property of CD * (K, N) spaces is that the disintegration of the given measure with respect to the radial distance function, can be represented by the one dimensional Lebesgue measure. This fact will be used in the proof of Lemma 2.13 which in turn is essential in the proof of the characterization theorem. The precise definition and the proof can be found in [10]. Proposition 2.6 (Disintegration of measure, Cavalletti-Sturm [10, Section 3]). Under the CD * (K, N) condition for K ∈ R, N ∈ (1, ∞), for fixed o ∈ X, we are able to disintegrate the given measure m by m = m r L 1 (dr), (2.4) where, m r is a Borel measure supported on the set {x ∈ X ; d( holds for any 0 < r ≤ R and x ∈ supp m. We denote by M C(·) the class of all normalised pointed metric measure spaces satisfying (2.5) for a given non-decreasing function C : (0, ∞) → (0, ∞). We have the following compactness and metrizability theorem.
For a given pointed metric measure space (X, d, m, x) with x ∈ supp m and r ∈ (0, 1), we associate the rescaled and normalised pointed metric measure space (X, d r , m x r , x), where d r := d/r and, There exists a non-decreasing function C : (0, ∞) → (0, ∞) depending only on K, N such that all RCD * (K, N) spaces belong to M C(·) (for instance, see Sturm [37]). Hence for RCD * (K, N) spaces, convergence with respect to D C(·) and that with respect to the pointed measured Gromov-Hausdorff topology coincide. Theorem 2.9 ( [24]). Let K ∈ R and N ∈ (1, ∞). Then the class of normalized RCD * (K, N) pointed metric measure spaces is closed (and therefore compact) with respect to D C(·) .
It is easy to see that for any λ > 0, (X, λd, m) satisfies the RCD * (λ −2 K, N) condition provided that (X, d, m) is an RCD * (K, N) space. This will imply that Tan(X, d, m, x) consists of RCD * (0, N) spaces for any point x ∈ supp m.
Therefore we are able to find a pointỹ ∈ Im (γ 1 ) and a small number η > 0 such that This is a contradiction. Then, if L 1 (I) > 0, there exists a closed interval I ′ with L 1 (I ′ ) > 0 such that L 1 (I ′ \ I) = 0.
Proof. We will use the regularity of the Lebesgue measure along with the Measure Contraction Property to find such a closed interval. By the regularity of the Lebesgue measure, for any ǫ > 0, one can find a closed set C and an open set U with C ⊂ I ⊂ U and such that L 1 (U \ C) < ǫ. First of all, this means that we can assume I is closed (otherwise, replace it with C and notice that C has positive measure for ǫ small enough). Claim 2.15 below shows that the measure contraction property implies that the set, I, is invariant under dilations (in a suitable sense that will be made clear in below). Let r(·) := d(x, ·) be the distance function from x.
Proof. Let (X, d, m) be an RCD * (K, N) space for K ∈ R, N ∈ (1, ∞). Take two distinct points x and y with d(x, y) = l. We denote a geodesic connecting x to y by γ 1 . Let r 0 > 0 be a positive number such that B r 0 (y) ∩ B r 0 (x) = ∅. We disintegrate m with respect to the distance function, r(·) := d(x, ·), that is, For J ⊂ R and V ⊂ X, let V J := {w ∈ V ; d(x, w) ∈ J}. Note that for any measurable subset V ⊂ X, if m r (V) > 0 for a.e. r ∈ J with L 1 (J) > 0 then, m(V J ) > 0 and obviously, m(V) > 0. Now, let I be the measurable subset defined by (2.6) and assume L 1 (I) > 0. Suppose a measurable subset J ⊂ (l − r 0 , l) with L 1 (J) > 0 satisfies m(A J ) > 0. Let τ ∈ (0, 1) be a number for which, L 1 ((τJ) \ I) = 0 and A τ J = ∅. Without loss of generality, we may assume m r (A J ) = m r (A) > 0 for all r ∈ J.
Let π ∈ OptGeo(µ, δ x ), where µ := χ A J m/(m(A J )) ∈ P (X). Note that by construction, µ ≪ m. Hence we are able to find a map T t : X → X such that (T t ) * µ = µ t = (e t ) * π, which is a geodesic from µ to δ x (see Gigli-Rajala-Sturm [25, Theorem 1.1]). Since L 1 ((τJ) \ I) = 0 (i.e. τ J is a subset of I in a.e. sense), we must have m r (A) = 0 for a.e. r ∈ τ J. Accordingly, Now, we consider two different cases: Case I: Suppose there exists a measurable subset B ⊂ A J with m(B) > 0 (hence (e 0 ) * π(B) = µ(B) = m(B)/m(A J ) > 0) such that for π-a.e. geodesic c w connecting w ∈ B to x, one has c w 1−τ ∈ A which readily implies that c w 1−τ ∈ A τ J . By the MCP condition, we have m(T 1−τ (B)) > 0. More precisely, since (e 0 ) * π(B) is positive, so is (e 1−τ ) * π(T 1−τ (B)). This means that m( Case II: Suppose for any measurable subset B ⊂ A J with m(B) > 0 one has for π-a.e. geodesic c w connecting w ∈ B to x, c w 1−τ ∈ A c . This implies that for π-a.e. c ∈ Geo(X), one has c 1−τ ∈ A c . Recall that we denote the geodesic connecting from x to y by γ 1 . We claim that there exists s 0 > 0 such that γ 1 This is true since by the Lebesgue density theorem, at almost every point t ∈ τ J, one has Hence, taking a Lebesgue point t ∈ τ J greater than inf τ J, and repeating the above Without loss of generality, we may assume that t 0 = d(x, γ 1 t 0 ) ∈ J is a Lebesgue point in J (otherwise, one can repeat the above arguments by replacing J with its Lebesgue points). Let ξ > 0 be a positive number such that Therefore, we obtain which is a contradiction.

Claim 2.16 (invariance of I under dilations).
For any 0 < t ≤ 1 and up to a set of measure zero, one has In other words, inside the interval (l − r 0 , l), I is invariant under dilations. In particular, for t ≪ 1, we get t −1 I ∩ (l − r 0 , l) = ∅ ⊂ I and for t = 1, we have which means t ′ (l − r 0 , l) ∩ I ⊂ I, which is obviously a contradiction. Now to finish the proof of the Claim 2.14, suppose, s ∈ I is a Lebesgue density point of I. This means For any ǫ > 0, choose δ > 0 such that Then for any t ≥ 1, using the scaling property of the Lebesgue measure, and the scale invariance of I, we can compute indeed, by the invariance of I under dilations, we have tI ∩ (l − r 0 , l) ⊂ I and this then would imply that Hence, In above, the last inequality follows from the definition of k since, Therefore, first letting δ → 0, and then ǫ → 0 in (2.8), we get This argument can be applied to any Lebesgue density point, s, in I (and we know almost every point of I is so). So, with a little bit more work, one can in fact prove that if s 0 := inf I, then Remark 2.17. The conclusion of Claim 2.14 is obviously wrong for arbitrary metric measure spaces (one needs MCP or some sort of curvature conditions. ). The following is a counterexample: Let C ⊂ [0, 1] be a closed nowhere dense Cantor set with positive Lebesgue measure (such sets exist). Take the isometric product Remark 2.18. One can weaken the assumptions in Lemma 2.13. It is not essential to assume d(x, y) = d(x, z). We just need two sets A and B that are included in {w ∈ X ; r 1 ≤ d(x, w) ≤ r 2 } for a pair of numbers 0 < r 1 < r 2 < ∞.

PROOF OF THE CHARACTERIZATION THEOREM
Let (X, d, m) be a metric measure space. Then, the RCD * (K, N) condition for K ∈ R and N ∈ (1, ∞), or more precisely, the locally doubling condition will imply that m satisfies m(U) > 0 for any open set U ⊂ supp m. For brevity, when there is no confusion, we will denote Tan(X, d, m, x) by just Tan(X, x). Definition 3.1. We define the following subsets of X based on the point-wise structure of the tangent space: [32]).
. Let x ∈ X be a point and suppose γ is a geodesic joining two points p, q ∈ X \ {x} that also passes through x. Suppose there exists a point z / ∈ Im (γ) with d(z, x) = d (z, Im (γ)). Then, there exists a pointed proper geodesic metric measure space, (W, d W , m W , w), with diam W > 0 such that R × W ∈ Tan(X, x). In fact, every tangent is of the form, R × W with diam W ≥ 0 and W depending on the tangent.
where t n is the infimum of the numbers t such that η(t) ∈ B 1/n (x). Then obviously, z n ∈ ∂B 1/n (x). Set w n := η(t n + (d(z, x) − t n )/2) and notice that d(x, w n ) + d(w n , z n ) = d(x, z n ) holds for any n ∈ N. Denote by d n , the normalized metric d/n. A simple calculation using the local doubling property implies So, there exists a positive constant C > 0 such that m x n (B d n 1/2 (w n )) ≥ C for any n ∈ N, where, m x n is the normalized measure with respect to d n at x. Thus, in the virtue of the splitting theorem, we deduce that a subsequence of the pointed normalized metric measure spaces (X, d n , m x n , x) converges to a product space Proof. Suppose not. Then, by the definition, for x ∈ W E 1 , there exists a proper metric measure space (W, w) with diam W > 0 such that (R × W, (0, w)) ∈ Tan(X, x). The stability of RCD * condition under D C(·) implies that R × W is an RCD * (0, N) space. The splitting theorem then implies that W is one point (see Theorem 2.10). This is in contradiction with the assumptions on W.
Proof. The proof is similar to the proof of Proposition 4.1 in Honda [27]. Suppose that there exists a point x ∈ R 1 such that x is not an interior point on a geodesic.
Proof. First of all, rescaling the metric if necessary, we may assume that Diam X > 1. By Theorem 4.1 in [32], there exists a number β = β(N) > 2 with the following property: there exists a large numberR i ≫ 1 such that for any R ≥R i there exist . Take points p i , q i ∈ X with the following properties: (1) p i , q i ∈ Im (c i ) and p i , q i are on opposite sides of ξ i , Notice that one can always find such points p i and q i on the geodesic c i since, This is what we wanted to prove in this claim.
This means that s i → 0 as i → ∞. Using the pre-compactness, a subsequence (X, . Now, our construction implies that there exist a limit point, z ∈ ∂B 1 (y), corresponding to a sequence of points, The following theorem is key.
By the assumption, we are able to take y n ∈ B η/n 2 (x) \ Im (γ) and z n ∈ Im (γ) so that d(z n , y n ) = d(y n , Im (γ)). By Lemma 3.2, z n = x for n large enough. We may assume z n ∈ P := {γ t ; t > 0}. Now take w n ∈ N := {γ t ; t < 0} so that Set l n := d(z n , y n ). Then by using the doubling property (also see the proof of Lemma 3.2), we have m(B l n /2 (y n )) > 0 and B l n /2 (y n ) ∩ Im (γ) = ∅. Let θ be a unit speed geodesic from y n to z n and set α n := 1/2 min{d(x, z n ), l n /n 2 }.
Then, for some k ≥ 2, l n > η/kn 2 and θ(l n − η/kn 2 ) ∈ B α n (z n ) \ Im (γ) (otherwise, one can find a point on γ that is strictly closer to y n than z n is). Therefore, B α n (z n ) \ Im (γ) is non-empty. And by the doubling property, Claim 3.8. There exists a point x n ∈ B α n (z n ) and a geodesic c n : [0, 1] → X connecting w n and x n such that d(x, Im (c n )) > 0.

Proof of Claim.
To prove the claim, we are going to use an argument similar to the one in Rajala-Sturm [36]. From Rajala-Sturm [36], we know that the optimal transport between any two absolutely continuous measures in a space satisfying strong CD(K, ∞) condition is concentrated on nonbranching geodesics. The idea of the proof is that if there exists a π ∈ OptGeo(µ 1 , µ 0 ) (for absolutely continuous µ i ∈ P 2 (X)) that does not live on non-branching geodesics, then via restriction in time and space and using disintegration, one can find a measure M on Geo(X) × Geo(X) and a family of geodesic pairs Γ a ⊂ Geo(X) × Geo(X) (a ∈ (0, 1)) with M(Γ a ) > 0 and m (e 0 (p 1 (Γ a ))) m (e 0 (p 2 (Γ a ))) > 0, where, the geodesic pairs in Γ a satisfy the following conditions: there exists a sufficiently small ξ > 0 such that restr a 0 γ 1 = restr a 0 γ 2 and restr a+ξ 0 γ 1 = restr a+ξ 0 γ 2 for any (γ 1 , γ 2 ) ∈ Γ a . Then, writing down the K−convexity conditions for the entropy of the transportation from e 0 (p 1 (Γ)) ∪ e 0 (p 2 (Γ)) to e 1 (p 1 (Γ)) ∪ e 1 (p 2 (Γ)), one proves that the underlying space fails to satisfy the strong CD(K, ∞) condition.
To prove Claim 3.8, we are going to prove that the assumption that every geodesic connecting w n to a point x n ∈ B α n (z n ) passes through x and the fact that m (B α n (z n ) \ Im (γ)) > 0 would provide us with such family of "bad" geodesics and that would lead to a contradiction.
Suppose for w n ∈ N ∩ B η/n 2 (x) and for any geodesic c n connecting w n to a point x n ∈ B α n (z n ), there exists a time t ∈ (0, 1) such that c n (t) = x. Consider µ 0 := δ w n and µ 1 := χ B αn (z n ) m/m(B α n (z n )). By Theorem 1.1 and Corollary 1.6 in Gigli-Rajala-Sturm [25], one could find a uniqueπ ∈ OptGeo(µ 1 , µ 0 ) that is induced by a map. The optimal planπ also satisfies (e t ) ♯π ≪ m for any t ∈ [0, 1). Define a map σ : Geo(X) → Geo(X) by σ(γ) t := γ 1−t . Let π denote the measure σ ♯π . Then, π satisfies µ t := (e t ) ♯ π ≪ m for any t ∈ (0, 1] and µ t is a geodesic connecting µ 0 to µ 1 . Note that π is supported on the branching subset Γ ⊂ Geo(X) of geodesics starting off as γ. x). Thus by the inclusion relation, g(B) ⊂ g (B α n (z n ) \ Im (γ)) holds. Now, by assumption we know that for almost every geodesic θ in the support of π, there is a time, t θ , such that θ(t θ ) = x. We replace θ, up to time t θ , by γ| [0,t θ ] . Also, notice that, for this family of geodesics, the branching time parameters, a and ξ can be chosen as follows: Therefore, by the uniqueness of π, and from the proof of Lemma 2.13 and the above argument, we deduce that there exist two subsets Γ 1 , Γ 2 ⊂ supp π with π(Γ 1 )π(Γ 2 ) > 0 such that for any γ 1 ∈ Γ 1 , there exists γ 2 ∈ Γ 2 with restr a 0 γ 1 = restr a 0 γ 2 and γ 1 (1) ∈ B, γ 2 (1) ∈ B α n (z n ) \ Im (γ). By restricting and rescaling π, we obtain a restricted plan π that is supported on branching geodesics (with the abuse of notation, we will also denote this restricted measure by the same character , π). Now, we have at our disposal, all the ingredients needed for the arguments in Rajala-Sturm [36] to work. So, employing the exact same arguments as in Rajala-Sturm [36], one obtains two measures π u , π d with the following properties : (a) β := π u (Geo(X)) = π d (Geo(X)).
(c) For fixed small number b > 0, there exists a positive number C > 0 such that Exploiting the K-convexity of the entropy along the plan (π u + π d )/(2β) from b to a + ξ (in a similar fashion as in Step 7 in [36]), we will get a contradiction. See the Appendix A for detailed computations.
The proof of Claim 3.8, in fact, implies that for m-a.e. x n ∈ B α n (z n ) and for π-almost every geodesic θ, connecting w n to the point x n ∈ B α n (z n ), we know θ does not pass through x. Thus we find the family of geodesics, {c n } n∈N , from w n to a point x n ∈ B α n (z n ) with d(x, Im (c n )) > 0.
Moreover, we may assume that π-a.e. geodesics, c n do not intersect P since, otherwise, one could replace the geodesic c n that intersect P with the geodesics,c n given bỹ Now, the collection ofc n 's would form a family of geodesics from w n to x n of positive π-measure and passing through x, this is in contradiction with the uniqueness of π and the proof of Claim 3.8. So far, we have that π-almost every geodesic does not pass through x and does not intersect P. Pick one of these good geodesics c n .
Let L n denote the distances d(x, w n ) = d(x, z n ). We get Let us consider the rescaled metric measure space (X, d L n , m x L n , x). Since x ∈ R 1 , we have X L n → R (taking subsequence if necessary). Let f n : X L n → R be the approximation maps that realize the convergence X L n → R. Since (X, d L n , m x L n , x) → (R, d E , L 1 , 0), there exist points on each Im (c n ) that converge to 0 ∈ R and consequently any sequence of points, c n (t n ) with d (x, c n (t n )) = d (x, Im (c n )) also has to converge to x. Thus, we are able to find a sequence t n such that c n (t n ) satisfies d(x, c n (t n )) = d(x, Im (c n )) and f n (c n (t n )) → 0 ∈ R. Indeed, every point c n (t) obviously satisfies Therefore, lim sup d L n (x, c n (t)) < ∞. This means that the image of the geodesic c n approaches to Im (γ) as n → ∞ in the L n -scale. Also since d L n (x, w n ) = d L n (x, z n ) = 1, c n (t n ) does not go closer to neither w n nor z n . Define s n := d(x, c n (t n )) and consider (X, d s n , m x s n , x). If lim inf n→∞ d s n (Im (γ), c n (t n )) > 0, we find a point in the limit space that is not on the geodesic corresponding to Im (γ). This is a contradiction to x ∈ R 1 .
This means points c n (t n ) are converging to a point on γ in the s n −scale. Assume that c n (t n ) converges to a point in P in the s n -scale (the case, c n (t n ) converging to a point in N in the s n -scale can be ruled out in a similar fashion). Pick times t ′ n such that t ′ n ≤ t n and d(c n (t ′ n ), c n (t n )) = s n . It is easy to see that we can find such a point c n (t ′ n ) since the assumption that c n (t n ) converges to a point in P implies d(w n , c n (t n )) > s n for n large enough. By the construction, d(x, c n (t ′ n )) ≥ d(x, c n (t n )) = s n . Hence d s n (x, c n (t ′ n )) ≥ 1. Since x ∈ R 1 and d s n (c n (t ′ n ), c n (t n )) = 1, c n (t ′ n ) converges to a point on Im (γ) in the s n -scale.
Let a := lim n h n (c n (t ′ n )) ∈ R, where h n : X s n → R are approximation maps. Since d s n (c n (t ′ n ), c n (t n )) = 1, a = 0 or a = 2. If a = 2, this contradicts the minimality of c n . Thus a = 0. Note that . Taking a subsequence if necessary, we know X K n → R via the approximation maps g n : X K n → R. Since x ∈ R 1 , d K n (x, c n (t ′ n )) = 1, and s n ≤ K n ≤ 2s n , d K n (c n (t ′ n ), Im (γ)) → 0 and g n (c n (t ′ n )) → −1 or 1 ∈ R. However, again by s n ≤ K n ≤ 2s n , d K n (x, c n (t ′ n )) ≤ 2d s n (x, c n (t ′ n )) → 0. This is a contradiction. Now, Consider pointed normalized metric measure spaces (X, s −1 n d, m x s n , x) that converge to (Y, d Y , m Y , y) ∈ Tan(X, x) in the measured Gromov-Hausdorff sense. By the rescaling, it is clear that (Y, y) is not isomorphic to (R, 0). This contradicts x ∈ R 1 .
(2) x ∈ X \ R 1 . Since R 1 = ∅, one can find a point y ∈ R 1 . By the proof of (1) above, a neighbourhood of any such y is isometric to an open interval. Therefore, R 1 is an open set. If R 1 is closed, then X must be R 1 itself. This contradicts the existence of x ∈ X \ R 1 . Note that R 1 is an open 1− dimensional manifold. If the open set R 1 is a circle, take a point, p in the circle that is the closest point from x, Lemma 3.2 implies that there exist a tangent cone at p that is not isometric to R. This is a contradiction (one can also see the contradiction by noticing that a circle is closed).
The maximal connected open subset in R 1 , which contains y ∈ R 1 , is a locally minimizing curve γ : (−a, b) → X, a, b ∈ (0, ∞], which satisfies γ 0 = y. Furthermore, γ −a := lim t→−a γ t and γ b := lim t→b γ t when a, b = ∞, do not belong to R 1 . Locally, a neighbourhood of each point in R 1 is isometric to (−ǫ, ǫ). This means the maximal connected subset in R 1 should be a local minimizing unit speed geodesic.
Just to make it more clear, we can argue as follows: Let γ : (−a, b) → R 1 ⊂ X be a locally minimizing curve with γ(0) = y ∈ R 1 . If p = γ(t 1 ) = γ(t 2 ) for some t 1 , t 2 ∈ (−a, b) and t 1 = t 2 , then since a neighbourhood of p ∈ R 1 is isometric to an interval, we deduce that γ has to be periodic (after trivially expanding its domain to R) so γ ⊂ R 1 is a circle. But as we previously showed, this can not happen. Therefore, from the argument above, we can assume γ has no self-intersections and can be extended from either end in a locally minimizing fashion as long as a (or b) stays finite. Suppose (−a, b) (a, b ∈ R ∪ {∞}) is the maximal domain for the locally minimizing curve γ. Then, if b < ∞ (respectively a < ∞), we must have γ b := lim t→b γ t ∈ R 1 (respectively γ −a := lim t→−a γ t ∈ R 1 ) since otherwise, one can extend γ further and in a locally minimizing fashion.
When both a and b are ∞, consider a point , z on γ with d(x, z) = d(γ, x). Then, Lemma 3.2 implies that z / ∈ R 1 which is a contradiction. Without loss of generality, we assume b < ∞. Consider a geodesic θ : I → X from x to a point z ∈ Im (γ) that satisfies d(x, z) = d (x, Im (γ)). If z = γ t for t ∈ (−a, b), we will get a contradiction by part (1) or by using Lemma 3.2 (in this case, there exists a tangent cone at z which is not R). Without loss of generality, we may assume that z = γ b / ∈ R 1 . Suppose x = z. Notice that for any η > 0, B η (z) \ (Im (γ) ∪ Im (θ)) = ∅ since, otherwise a neighbourhood of z would be isometric to an open interval. Indeed, B η (z) \ (Im (γ) ∪ Im (θ)) = ∅ implies that a neighbourhood of z is just the concatenation of two minimal geodesics, γ and θ; also every geodesic joining two points in B η ). This means z ∈ R 1 which we know is not the case. In particular, the above argument ensures us that if x = z, one must have B η (z) \ (Im (γ) ∪ Im (θ)) = ∅ for any η > 0. Take a point w ∈ B η (z) \ (Im (γ) ∪ Im (θ)) and consider a geodesic α from w to the point v ∈ Im (γ) From now on, we just repeat a similar argument as in the case (1). For the sake of completeness, we give an outline of the proof. Take a point z ′ ∈ R 1 , which is close enough to z. In order to apply the argument in (1), we may assume that By considering the optimal transportation between µ 0 := m| A /m(A) and µ 1 := δ y , we are able to find a curve c from y to a point in A not passing through z ′ . This means that there exists a point in This means that x has to be the end point (after taking the closure of the geodesic) of the geodesic, γ ⊂ R 1 . Hence B ǫ (x) is isometric to [0, ǫ) for sufficiently small ǫ > 0. Remark 3.9. We note that in the proof of Claim 3.8, the geodesics are not branching at the same time but they are all branching within a tiny time interval [a, a ′ ] the length of which going to zero as n → ∞ and that is enough to get a contradiction. Another possible approach would perhaps be to non-linearly contract the geodesics toward w n so that all branch at the same time and then use the measure contraction property to get a contradiction. The difficulty in this approach is that since all geodesics are of constant speed and parametrized on [0, 1], when we perform such a non-linear contraction, we will end up with a family of geodesics that branch at the same time but their end points will all be on a sphere with center w n . This contradicts the measure contraction property or the spherical Bishop-Gromov inequality in, for example, non-collapsed Ricci limit spaces of dimensions strictly larger than 1. But in the setting of RCD * (K, N) spaces, it is unclear to the authors how to derive a contradiction having a family of branching constant speed geodesics parametrized on [0, 1] (all branching at the same time) with end points on a sphere. To the best of authors' knowledge, a spherical Bishop-Gromov volume comparison or measure contraction property (i.e. a volume comparison or measure contraction property for the co-dimension 1 measures) is yet not available in this setting. Also notice that even in the simplest example of the letter "Y" space (the tripod), the geodesics emanating from one point on one branch and going to other two branches, once parametrized on [0, 1] , are branching at different times (depending on their lengths).
Remark 3.10. In Theorem 3.7, we have in fact proven the stronger fact that in any RCD * (K, N) metric measure space, R 1 is an open and convex (convexity follows from arguments in the proof of part (2)) subset. In Ricci limit spaces, the convexity of all the regular sets follow from the recent developments by Naber and Colding but to the best of our knowledge, in the metric measure setting, this is not known (at least for R k , k ≥ 2). Definition 3.11. Let (X, d) be a geodesic, proper complete separable metric space. A positive Radon measure µ on X is a reference measure (in the sense of Cavalletti-Mondino [8]) for (X, d) provided it is non-zero, and µ-a.e. z ∈ X there exists π z , which is a positive Radon measure on X × X, such that where p i : X × X → X is the natural i-th projection maps i = 1, 2 and The measure π z is called an inversion plan. lp(µ) is the set of all points z ∈ X that has an inversion plan π z . Proposition 3.12. Let (M, g) be a complete Riemannian manifold of dimension 1 and let d g , m g be the Riemannian distance function and the Riemannian volume measure associated with g (respectively). Let µ be a locally finite Borel measure on M satisfying RCD * (K, N) condition for K ∈ R and N ∈ (1, ∞). Assume that supp (µ) = M. Then, µ and m g are reference measures for (M, d g ) and µ ∼ m g , and µ = e −V m g for some locally integrable function V.
Proof. The fact that m g is a reference measure follows from Cavalletti-Mondino [8]. We present an argument as to why µ is also a reference measure. First of all, the measure µ is a Radon measure ( [20,Theorem 7.8]). Since (M, d g , µ) satisfies the RCD * (K, N) condition for K ∈ R, N ∈ (1, ∞), µ does not have atoms, that is, µ({x}) = 0 for any x ∈ M. Assume (M, d g ) is isometric to (R ≥0 , d E ) (the other cases can be dealt with in a similar way). First of all, by Proposition 3.4 in [8], we have µ ≪ m g . Take z ∈ R >0 and fix it.
Step 1. We find a family of compact sets K n ⊂ M, n ∈ N and bi-Lipschitz maps Φ n (the so called local inversion maps), such that, • For every x ∈ K n , there exists a unique constant speed minimal geodesic γ xz : [0, 1] → M from x to z, which can be extended to [0, 1 + 2/n] → M as a minimal geodesic, • The map Φ n : K n → M defined by Φ n (x) := γ xz (1 + 1/n) is bi-Lipschitz onto its image.
Step 2. Define a map Φ : Take the measure π z := (Id, Φ) ♯ µ. We claim that π z satisfies all the properties required in Definition 3.11. It is clear that (p 1 ) ♯ π z = µ and π z (X × X \ H(z)) = 0 by the construction. The last property (p 2 ) ♯ π z ≪ m g is proven as follows. Let E ∈ R ≥0 be a Lebesgue negligible set, that is, m g (E) = 0. Since m g is also Hausdorff measure, m g (φ(E)) = 0 for any bi-Lipschitz map φ : M → M (see for instance [4, Proposition 3.1.4]). Therefore, we obtain Notice that the last equality follows since the sets ,φ −1 n (E), are Lebesgue negligible sets and µ ≪ m g . Hence, µ is a reference measure for (M, d g ). The same proof shows that m g is also a reference measure for (M, d g ).
First of all, we notice that for a given bounded Borel set Ω ⊂ R, the integral of the negative part of V on Ω is finite. Indeed, decompose V into the positive and the negative parts V = (V) + − (V) − . Then, using, x ≤ e x , we get For k >> 0, take V k = min{V, k}. V k is integrable w.r.t H 1 and any other absolutely continuous measure. Fix a closed interval [a, b] and denote by H 1 even which is restricted on [a, b]. Claim 3.14. For any measure µ ∈ P ([a, b]) that is absolutely continuous with respect to H 1 and µ ∈ D(Ent(·|H 1 )), we have Proof. Note that the equivalence H 1 ∼ e −V H 1 and the integrability of the negative part of V imply µ ∈ D(Ent(·|e −V H 1 )). Since V k is integrable, we can write Let U N (r) := −Nr 1− 1 N defined on R ≥0 . Then, on R >0 , U N is negative valued, decreasing and convex. Let then, from Sturm [37] and the fact H 1 ([a, b]) < ∞, we know that for any ν ∈ P 2 ([a, b]) We can now compute hence we get the desired result.
So now, let µ := 1 b−a H 1 be the normalized Hausdorff measure on [a, b] and notice that we have V k dH 1 is increasing and bounded above. Hence by monotone convergence theorem and since Ent( Take two distinct Lebesgue points x 0 , x 1 ∈ [a, b] of V with respect to H 1 , that is, to assume Note that by the Lebesgue differentiation theorem, and by factoring in the(K, N)-convexity of the Entropy functional, we get It is easy to see that W 2 (µ r 0 , µ r 1 ) = d E (x 0 , x 1 ). Moreover, the measure µ r t can be written as Taking the limsup of (3.3) as r → 0, one gets In particular, holds if x t is a Lebesgue point of V. Consider the function W which is defined by where, the infimum in the second line, is taken over all sequences {y i } approaching to x. By the definition of W and by (3.4), we obtain The same argument above leads to the consequence.  In this section, we present a totally different proof of this result when m = 1 by taking advantage of the convexity of the potential function V. Moreover, our presented theorem is a bit stronger (see Remark 4.3). (X, d, m) be an RCD * (K, N) space for K > 0, N ∈ (1, ∞). Then (X, d) is not isometric to a circle with its standard metric (S 1 (r), d) for any r > 0.

Theorem 4.2. Let
Proof. Suppose (X, d, m) is isometric to (S 1 (r), d, m). For simplicity, we omit r > 0. By Theorem 1.1, we are able to write m = e −V vol S 1 for a (K, N)-convex function V. First to see where the contradiction comes from, we assume V ∈ C 2 (S 1 ). Then V satisfies the differential inequality V ′′ ≥ K − (V ′ ) 2 /N (see the equation (1.2) in [19]). Since K > 0, we have V ′′ > 0 at critical points. On the other hand, V has a maximal point x 0 ∈ S 1 since V is continuous and S 1 is compact. Therefore V ′′ (x 0 ) ≤ 0. This contradicts. Now for general case, we know that V is continuous (and in fact Lipschitz). Supposex is a maximal point for V. Take x 0 , which is a contradiction.
Remark 4.3. In Colding [16], sequences of n-dimensional closed Riemannian manifolds with Ric ≥ n − 1 are considered. Our Theorem 4.2 also applies to weighted Riemannian manifolds with boundary as long as RCD * (K, N) condition for K > 0 and N ∈ (1, ∞) is satisfied.

Bishop-Gromov type inequalities.
In this section, we prove useful Bishop-Gromov type inequalities for RCD * (K, N) spaces. and, Let S K,N (t) for N > 1, K ∈ R be the following: Bishop-Gromov type inequalities for boundary measures hold on Ricci limit spaces (see Honda [26]). The same is also true for RCD * (K, N) spaces.
Then, In the virtue of the Lévy-Gromov isoperimetric inequality for RCD * (K, N) spaces that is proven in Cavalletti-Mondino [9], one gets where, I K,N,D (·) is the Milman's model isoperimetric profile (see Cavalletti-Mondino [9] and Milman [31] for the precise definitions). Our Theorem 5.2, in contrast to the Lévy-Gromov isoperimetric inequality, provides an upper bound for the surface measure m −1 (∂B t (x 0 )) in terms of m (B t (x 0 )). Notice that the two surface measures, m + and m −1 are a priori different but comparable in one direction on spheres. Indeed, by (5.3), we have Since A direct consequence of the inequality (5.1) is the following. holds for any s ∈ (0, 1]. Moreover, Proof. Once we prove (5.6), (5.7) will directly follow by using Theorem 5.2. Fix y ∈ X and R > 0. Take . Thus, repeating the same calculation as in the proof of Theorem 5.2, we write Upon letting δ → 0, we obtain m(B t (y)). (5.8) Notice that, these calculations actually imply that the small scale volume growth at any point is at most linear so we can write m(B t (y)) ≤ Ct for some C > 0. Also notice that Therefore, the RHS of (5.8) is bounded by Hence, holds for small δ > 0. The inequality (5.2) and the proof of Theorem 5.2 give the conclusion.
is a tangent cone at x ∈ X. Then, Proof. It is implicit in the splitting theorem applied to , in the measured Gromov-Hausdorff sense. In the virtue of Corollary 5.6, m W (B r (w)) ≤ Cr.
Lemma 5.8. For given r > 0, the function x → m(B r (x))/r is locally Lipschitz and in particular, locally uniformly continuous for r > 0.
Proof. A similar argument as in the proof of Theorem 5.2 can be applied here too ( also see Lemma 3.1 in [29]). For the reader's convenience, we give a proof. The notations below are as in the proof of Theorem 5.2. Fix a point x ∈ X. Take another point y ∈ X. For simplicity, set d : = d(x, y). Take a midpoint z ∈ X, that is, Therefore, for small d. Interchanging the role of x and y, gives (5.10) The right-hand side in (5.10) is independent of the choice of x, so using Corollary 5.4, we have the conclusion.
Remark 5.9. In (5.10), we have F ′ (r)/F(r) → ∞ as r → 0 and therefore, it does not tell us anything about the modulus of continuity of m(B r (x)) r . If we, a priori, assume the uniform continuity for r ≥ 0, we can prove that Proof. Suppose not. Let x ∈ M 1 \ M 1 . Hence, there exists a constant C > 0 such that C ≤ lim inf r→0 m(B r (x))/r. Take a sequence y i ∈ M 1 converging to x. For sufficiently small r > 0, we have C/2 ≤ m(B r (x))/r. By Lemma 5.8, |m(B r (x)) − m(B r (y i ))| ≤ Cr/4 for large i.
for any small r. This contradicts y i ∈ M 1 .
Then (X, d, m) is isomorphic to one of the metric measure spaces given in Theorem 1.1.
We can generalize the statement of above propositions in the following way. Define M k := x ∈ X ; lim inf r→0 m(B r (x)) r k = 0 .
The closeness of M k can be proven just in the same way as in Proposition 5.10. Then we conjecture: Remark 5.13. The Conjecture 5.12 is deeply related to a relation between given measure m and Hausdorff measure on regular sets. We speculate that, (5.11) being true, would imply that m restricted to R k is an Ahlfors k-regular measure. (also see the related work by David [18]).
APPENDIX A. EXPLICIT DETAILS OF THE PROOF OF CLAIM 3.8 Here, we will show that the K− convexity of the entropy fails under the branching phenomenon (even when the branching time is not the same but rather within a short time interval) as in Claim 3.8. One should keep the tripod example in mind while reading these computations.
We will be using the same notations as in the Claim 3.8 and almost the same calculations as in [36].
As we observed in Claim 3.8, one obtains two mutually singular measures π u and π d (in the tripod space analogy, the superscripts d and u mean up and down referring to the plans supported on either the upper or lower branches).
The trick is to write the K− convexity of Entropy along the measure curve at "fixed" times, t = b (a very small positive number less than a), t = a and t = a ′ = a + ǫ and along the measure curves ρ d t β and ρ u t β at "fixed" times t = a, t = a ′ = a + ǫ and t = 1. Recall that all the branching is happening within the tiny time interval (a, a ′ ) and hence, these two measure curves coincide for times t ≤ a. The we can continue as follows The above is equivalent to Taking into account (A.1), We can approximate the entropy of π u /β at time a + ǫ by On combining the upper estimate (A.2) and the lower estimate (3.1), we obtain The right-hand side of (A.3) is strictly negative (think of b ց 0) while the lefthand side approaches to 0 as ǫ goes to 0 (recall that ǫ → 0 as n → ∞). This is a contradiction. It is easier (computation-wise) to get the contradiction using the Rényi entropy instead of the Shannon entropy as we demonstrate in below. For simplicity we assume β = 1 and K ≥ 0. For general K, one would need to also incorporate torsion coefficients in the K− convexity estimates and the contradiction will follow by letting ǫ → 0 and then, N → ∞ (notice that for any K, the torsion coefficients, σ N (t) converge to t as N → ∞). These computations are similar to those carried out in Rajala [35].
Now, on letting ǫ → 0, we get which is an obvious contradiction for any N.