Scalar curvature via local extent

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.


Introduction
It is well known that the volume growth of the geodesic balls of an n-dimensional Riemannian manifold (M, g) is tightly related to curvature. On the one hand, the first non-trivial term of the asymptotic expansion of the volumes of infinitesimally small balls centered at a point x ∈ M is given, up to a constant, by the scalar curvature at x. Namely one has (1) Vol g (B M ε (x)) = Vol(B R n ε ) 1 − Scal g (x) n + 2 6 ε 2 + o(ε 2 ) , see for instance [Gra73]. On the other hand, thanks to the celebrated Bishop-Gromov volumes comparison theorem, complete manifolds with globally lower bounded Ricci (or sectional) curvature, enjoy an upper control on the volume growth of their geodesic balls, i.e. for all x ∈ M and r ≥ 0 Ric g ≥ (n − 1)K ⇒ Vol g (B M r (x)) ≤ Vol(B S n K r (x)), S n K being the simply connected n-dimensional space forms of constant sectional curvature K. Moreover, the equality in Bishop-Gromov is realized if and only if B M r (x) and B S n K r (x) are isometric. This latter characterization of the equality case can be seen as an extremal result for the recognition problem. Namely, a recognition Date: February 25, 2022. 1 problem in Riemannian geometry "asks for the identification of an unknown riemannian manifold via measurements of metric invariants on the manifold", [GM95]. In their program intended to attack extremal recognition problems, K. Grove and S. Markvorsen introduced a new metric invariant of a Riemannian manifold X (or of a more general metric space), which they called the q-extent. This roughly measures how far q-points of the space X can be one from the others. Namely, one defines xt q (X) := q 2 −1 sup (x1,...,xq)∈X q 1≤i<j≤q where the supremum is obviously a maximum when X is compact. In this case, following [GM95], we call q-extender the set of q points realizing the q-extent. Note that the q-extent naturally generalizes the diameter (i.e. the 2-extent) of a metric space and it is intimately related to the q-th packing radius pack q X := 1 2 max x1,...xq min 1≤i<j≤q d X (x i , x j ); [GM95,GM92]. One of the purposes of Grove and Markvorsen's recognition program was to study the relation between the q-extent and global lower bound on the sectional curvature. A (particular case of) a result of [GM95] says that an n-dimensional, n ≥ 2, Alexandrov space X with sectional curvature (in the sense of Alexandrov) greater or equal than 1 satisfies (2) xt n+1 (X) ≤ xt n+1 (S n ), S n being the unit sphere of R n+1 with its canonical metric. Moreover, the equality is realized in (2) if and only if diam X = π, which, together with Toponogov's diameter sphere theorem, implies that X is necessarily isometric to S n provided it is a Riemannian manifold. 1 According to what said above, it is then natural to expect that, similar to the volume growth, the first nontrivial term in the asymptotic expansion of the (n + 1)extent of infinitesimally small geodesic balls centered at a point x ∈ M involves the scalar curvature at x. This is the content of the next result.
Corollary 2. Let (M, g) be a smooth n-dimensional Riemannian manifold and x ∈ int(M ). Then .
Here and onB X ε (x) denotes the closed metric ball {y ∈ X : d X (x, y) ≤ ǫ} of the metric space (X, d X ). With an abuse of notation, sometime we will writeB d ε (x) to specify the metric we are considering on a (fixed) space X.
The main ingredients in the proof of Theorem 1 are • An asymptotic formula for the distance function in geodesic normal coordinates, [Bre09,Gal88]. On the tangent space at x, one can consider two different distances between two vectors u and v, that is the Euclidean one |u−v| and the distance induced by the Riemannian metric g pulled-back via the exponential map, i.e. d g (exp x (u), exp x (v)). Their difference is given (at the first nontrivial order) by R g (u, v, v, u), where R g is the Riemann tensor of g; see Lemma 9 below. • According to [Lil75], the only (n + 1)-extenders in the n-dimensional Euclidean ball are the regular simplexes. Here we need a quantitative analysis characterizing the (n + 1)-simplexes which almost realize the extent; see Proposition 6 below. • By its very definition it turns out that the scalar curvature is (twice) the average of the sectional curvatures of all the planes spanned by an orthonormal frame. The same holds true if instead one considers all the planes spanned by the vertices of a regular (n + 1)-simplex; see Lemma 10. In view of possible applications to metric spaces, some pathological behavior can arise in the (n + 1)-extenders of geodesic balls. For instance, for thin 2-dimensional cones, the 3-extent of a metric ball around the vertex is realized by 3 points including the vertex itself. In some non-rigorous sense the extender's shape is thus discontinuous with respect to the width of the angle. For this reason, we are led to introduce a slightly modified object, which we call boundary (n + 1)-extent and which can be defined for a closed geodesic ballB M ǫ as As it is clear from the proof, the Riemannian characterization of the scalar curvature given in Theorem 1 and Corollary 2 holds without changes if one replaces in the formulas the (n + 1)-extent of the geodesic balls with their boundary (n + 1)-extent. In particular we have the following Theorem 3. Let (M, g) be a smooth n-dimensional Riemannian manifold and x ∈ int(M ). Then .
Clearly, as in Corollary 2, also the formula (1), as well as the well-known similar expansion for the area of the surface of small geodeisc balls, allows an alternative definition of scalar curvature at x ∈ M depending on the geometry of (arbitrarily small) neighborhood of x. However, the characterization of the scalar curvature expressed via the (n + 1)-extent has the peculiarity of depending explicitely on the sole distance function of M , and not on the whole Riemannian structure. By a theoretical point of view, this advantage is negligible since a) the distance function uniquely determines the Riemannian metric tensor, [Pal57], and b) whenever the volume measure of the underlying space is given by its n-dimensional Hausdorff measure, also the volume measure depends on the distance function. However, the explicit dependence given in (3) seems nontrivial. Moreover, this point of view suggests that (3) could be used to give a definition of scalar curvature (bounds) of a metric space.
In this direction, we will propose some possible approaches. First, (3) can be used as it is to introduce a notion of point-wise defined (dimensional) scalar curvature on a metric space (X, d). The asymptotic limit in (3) in general will not exist, however a lim inf (resp. lim sup) version can still be used to define spaces with lower (resp. upper) bounded scalar curvature, see Definition 14 and 15 below. This notion of metric scalar curvature reveals consistent at least on metric spaces with lower bounded curvature in the sense of Alexandrov (on which a unique natural concept of integer dimension is defined). In fact, we have the following result. The inequality part was already observed in [GM95, (7)], while the rigidity is proven in Proposition 17 below).
Theorem 4. Let (X, d) be a n-dimensional CBB(k) Alexandrov space, for some k ∈ R. Then for all x ∈ X and ǫ > 0, where S n k is the simply connected n-dimensional space form of constant curvature k. Moreover, if ǫ ≤ π 4 √ k or k ≤ 0, then equality holds in (5) if and only ifB X ε (x) contains an isometric copy of ∆ n+1 k (ǫ) with totally geodesic interior. Here, ∆ n+1 k (ǫ) is the unique (up to isometries) regular (n + 1)-simplex inscribed inB S n k ε .
A special class of two dimensional metric spaces is given by the surfaces with bounded integral curvature introduced by Alexandrov. These spaces have a natural notion of curvature measure, which can be described for instance as a weak limit of integral curvatures along an approximating sequence of smooth Riemannian metrics (for details, see the references given in Section 4). In Proposition 20 we will show that the regular part of this curvature measure can be point-wisely obtained via the local extent as in (3). Clearly on the support of the singular part of the curvature measure, the 3-extent of geodesic ε-balls does not converge as ε → 0, so that a point-wise definition is there ill-posed.
Because of the intrinsic singularity of the underlying spaces, it would be more convenient to define on a metric space a scalar curvature measure instead of a point-wise definition. I'm grateful to J. Bertrand and M. Gromov for pointing this out to me. In the interesting paper [KLP17], Kapovitch, Lytchak and Petrunin proposed a notion of mm-curvature measure generalizing the asymptotic characterization (1). In particular they proved that such a measure is well-defined on BIC surfaces, although unfortunately it does not coincide with the intrinsic curvature measure. In Section 5 we will propose a construction similar to that of [KLP17], generalizing to measures on non-smooth surfaces the expansions of the extent and of the boundary extent of geodesic balls (formulas (14) and (15)) instead of the expansion of the volumes (1). Namely, consider a smooth n-dimensional Riemannian manifold (M, g). For some r 0 > 0 small enough define two families of measure {e r } 0<r<r0 and {∂e r } 0<r<r0 by In view of Theorem 1 and 3 we have that both r −2 e r and r −2 ∂e r converge weakly in the sense of measure to Scal g ·H n . In the non-smooth case, we can prove that on a surface with bounded integral curvature the families {r −2 e r } and {r −2 ∂e r } are uniformly bounded, so that at least some subsequence converges; see Theorem 21. The paper is organized as follows. In Section 2 we prove the Riemannian characterization given by Theorem 1 and propose a possible generalization involving the p powers of the distances, see Theorem 12. In Section 3 we consider the point-wise definition of (bounds on the) scalar curvature on Alexandrov spaces. In the last two sections we focus on surfaces with bounded integral curvature. In Section 4 we show that the point-wise characterization holds on the regular part of BIC surfaces, while in Section 5 we introduce the (boundary) extent curvature measures inspired by the construction of [KLP17].

Proof of the Riemannian characterization
Before starting the proof of Theorem 1, let us remark that the result is trivial for M = R n . In this case, the value of xt n+1 (B R n 1 ) is given by the following lemma; see [Lil75, Lemma 3].
Moreover the regular (n + 1)-simplexes inscribed inB R n 1 are the only (n + 1)-tuples of points realizing the equality.
We recall here some basic useful facts concerning regular simplexes in the Euclidean space. Fix {e i } n i=1 the canonical orthonormal basis of R n . For p = 1, . . . , n and for k = 1, . . . , p + 1, we introduce the vectors S p k ∈ R n defined by S p 1 = e p for p = 1, . . . , n and by the inductive rule It is easy to see that In the following, we will need a stable version of the equality case in Lemma 5, that is, simplexes which approximate the equality are almost regular. From now on, following standard notation, we will say that a (possibly vector-valued) function f : (0, ε 0 ) → R for some ε 0 > 0 satisfies f (ε) = O(ε a ) for some a ∈ Z if there exists a constant C (possibly depending on the underlying manifold M ) such that lim sup ε→0 |f (ε)|ε −a ≤ C.
As announced above, we have the following Proof. Let G be the barycenter of the given points, i.e. G = 1 n+1 n+1 k=1 P ε,k . Reasoning as in [Lil75, Lemma 3], we get where α is the angle at the origin in R n(n+1)/2 formed by the vector E = (|P ε,k − P ε,j |) 1≤k<j≤n+1 ∈ R n(n+1)/2 and by the vector I ∈ R n(n+1)/2 all whose components are 1. Condition (8) ensures that cos α = 0 for ε small enough and that n(n + 1) 2 2 (n + 1) sin 2 α + cos 2 α (n + 1) Let ν ∈ R be the constant such that νI is the projection of E onto the line spanned by I.
We introduce vectors P p ε,k ∈ R n for p = 1, . . . , n and k = 1, . . . , p + 1 which are defined as follows. For p = n we set P n ε,k := P ε,k for k = 1, . . . , n + 1, while for p < n they are defined inductively on p by the same recursive relation as in (6), that is, Lemma 7. With notations above, we have that Moreover for all 1 ≤ k < p ≤ n and for all 1 ≤ j ≤ k + 1 it holds and where ·, · is the standard scalar product of R n .
Lemma 8. For every 1 ≤ p ≤ n, there exists an isometry In particular (19) is satisfied for p = 1. Now, suppose that (19) is satisfied for some 1 ≤ p < n. Composing A p with a suitable further isometry A ′ p , we can find a new isometry A p+1 ∈ O(n) such that (19) is satisfied for p + 1 instead of p. Namely, recall that the vectors A p S q j , with 1 ≤ q ≤ p and 1 ≤ j ≤ q + 1 are all contained in a p-dimensional hyperplane H p of R n . Moreover A p S p+1 1 ∈ H ⊥ p < R n . Then one can take the isometry ) and such the projection of P p+1 ε,1 onto the (n − p)-dimensional hyperplane H ⊥ p < R n is parallel to . Explicitly one has that for each ε there exists A ′ p ∈ O(n) and a unique positive α such that According to (19), using also (13) and (14), we deduce Again by (14) we have also that α = α(ε) = 1 + O(ε). Setting A p+1 := A ′ p A p , we have proved (19) for p + 1, hence recursively for every 1 ≤ p ≤ n.
We now come back to the proof of the main theorem.
Proof. This follows easily from the asymptotic formula which can be proved via explicit computations; see for instance [Gal88,Bre09].
for some function A : (0, ε 0 ) → O(n) taking values in the isometries group ofB R n 1 . We are going to use the following Lemma, which will be proved later.
It remains to prove Lemma 10.
Proof (of Lemma 10). For p = 1, . . . , n and k = 1, . . . , p + 1, define unitary vectors S p k ∈ R n as in (6), and fix an orthonormal basis of T x0 M = R n by setting e j = S j 1 for 1 ≤ j ≤ n.
We are going to prove by induction on n that 1≤k<j≤n+1 R x0 (S n k , S n j , S n j , S n k ) = (n + 1) 2 n 2 1≤k<j≤n Sect(x 0 )(e k ∧ e j ).
For the shortness of notation, for vectors X, Y ∈ R n , we set κ(X, Y ) := R x0 (X, Y, Y, X).
Inserting (32)  A possible generalization. For real positive p, one could also define the p th order q-extent of a metric space (X, d) as It turns out that for 1 ≤ p < 2, Theorem 1 and Corollary 2 generalize to xt (p) q (X). Namely one has Theorem 12. Let (M, g) be a smooth n-dimensional Riemannian manifold and x ∈ int(M ). Then Corollary 13. Let (M, g) be a smooth n-dimensional Riemannian manifold and x ∈ int(M ). Then .
These results can be proved essentially as the case p = 1 treated above. The main difference is the proof of the stability result stated as Proposition 6. Namely, one has to replace relation (9) with the inequality 1≤k<j≤n+1 |P ε,k − P ε,j | p ≤ n(n + 1) 2 This latter can be obtained using a quantitative version of Hölder inequality; see for instance [Ald08, Theorem 2.2].
Note that, for p ≥ 2, regular (n + 1)-simplexes are no more the (unique) (n + 1)extenders of Euclidean balls, so that the proof fails to work in this case.

Local extent in metric spaces
The curvature of metric spaces is the object of an active research field. Since the seminal work by Alexandrov, a notion of metric sectional curvature lower bounded is provided. For later purposes, we recall that a locally compact metric space (X, d), whose metric d is intrinsic, is a space of curvature bounded above (resp. below) by k in the sense of Alexandrov if in some neighborhood of each point the following holds: For every ∆abc and every point d ∈ [ac], one has |db| ≤ |db| (resp.|db| ≥ |db|) wherē d is the point on the side [āc] of a comparison triangle ∆ābc such that |ad| = |ād|. Here a comparision triangle is a triangle of verticesā,b andc in the k-plane S 2 k (i.e. the 2-dimensional simply connected space of constant curvature k) satisfying |ab| = |āb|, |ac| = |āc| and |bc| = |bc|. Other equivalent definitions can be found for instance in [BBI01].
In the last decades several possible notions of Ricci curvature bounds for (measured) metric spaced have been proposed and successfully investigated.
On the other hand, the aim for a metric notion of scalar curvature is much more recent and seems more difficultous; see for instance [Gro14,Sor16,Gro17] and references therein. Specific answers to this problem have been given in particular context; see for instance [Mac98,KLP17] for Alexandrov surfaces and [Ber02,Ber03] for Alexandrov definable sets in o-minimal structures.
Here we propose the following definition Definition 14. Let (X, d) be a locally compact metric space and x ∈ int(X). We define the (n-dimensional) scalar curvature of X at x as , whenever the limit exists.
Note that, in general, the scalar curvature of a metric space is not point-wise defined (think for instance to the singular part of a polyhedral space). Accordingly, it could be interesting to consider bounds on the scalar curvaure.
Definition 15. Let (X, d) be a locally compact metric space. We say that X has (n-dimensional) scalar curvature greater or equal than k ∈ R at x ∈ X, and we write Scal Similarly, we say that X has (n-dimensional) scalar curvature smaller or equal than k ∈ R at x ∈ X, and we write Scal According to Lemma 5 below, in case k = 0 the definition above specifies as follows. A metric space X has nonnegative scalar curvature at x if lim sup ε→0 ε −1 xt n+1 (B X ε (x)) ≤ 2 n + 1 n , while X has nonpositive scalar curvature at x if lim inf ε→0 ε −1 xt n+1 (B X ε (x)) ≥ 2 n + 1 n .
Thanks to Theorem 1 and the subsequent discussion, the above definitions of (n-dimensional) scalar curvature are clearly consistent with the classical one when the underlying space is a Riemannian manifold. Moreover, they are also consistent with the curvature bounds in the sense of Alexandrov.
Recall that (finite dimensional) Alexandrov spaces with a lower curvature bound have a natural notion of dimension (in fact, all the reasonable notions of dimension coincide, and the dimension is a positive integer; see [BBI01, Chapter 10]). Accordingly, as in the Riemannian setting one expects the following result, which is in fact a direct consequence of [BBI01, Proposition 10.6.10].
Theorem 16. Let (X, d) be a n-dimensional CBB(k) Alexandrov space, for some k ∈ R, then for all x ∈ X and ǫ > 0, where S n k is the simply connected n-dimensional space form of constant curvature k. In particular Let ∆ n+1 k (ǫ) be the unique (up to isometries) regular (n + 1)-simplex inscribed inB S n k ε (that is, with totally geodesic faces in S n k ). Note that ∆ n+1

0
(1) = S n+1 . Similarly to the analogous rigidity result for the packing radius (see [GM95,Lemma 3.3]), we have the following Proposition 17. Let (X, d) be a n-dimensional CBB(k) Alexandrov space, for some k ∈ R. Suppose that for some x ∈ X and ǫ > 0 (with ǫ ≤ π 4 √ k if k > 0), Then an isometric copy of ∆ n+1 k (ǫ) with totally geodesic interior is inscribed in B X ε (x). In particular, there exists δ 0 depending on ǫ, k and n such thatB X δ (x)) is isometric toB S n k δ for every 0 < δ ≤ δ 0 .
Remark 18. As for [GM95, Lemma 3.3], Proposition 17 is sharp, in the sense that in general we can not expectB X ε (x) andB S n k ε to be isometric. A trivial counterexample is given by the Alexandrov space ∆ n+1 k (ǫ), endowed with the metric induced by S n k . The bound on ǫ is given by the fact that regular simplexes are no more extenders ofB S n k ε for ǫ close to π 2 √ k (although they continue being packers).
Proof. Let {Q j } n+1 j=1 be a (n + 1)-extender forB X ε (x). Let f : X → S n k be the noncontracting map given by [BBI01, Proposition 10.6.10], and define {P j } n+1 j=1 ⊂ B S n k ε (x) as P j = f (Q j ). By the non-contractivity of f , it holds for every k and j. In particular Equality in (36), together with (37), implies that (38) d S n k (P j , P k ) = d(Q j , Q k ) for every k and j and that {P j } n+1 j=1 is a (n + 1)-extender forB S n k ε , which is hence given by ∆ n+1 k (ǫ); see [GM95,page 9]. According to the characterization given in [GM95,page 19], {P j } n+1 j=1 is also a (n + 1)-packer ofB S n k ε . From (38), we deduce that pack n+1 (B X ε ) ≥ pack n+1 (B S n k ε ). By angle comparison, this inequality is in fact an equality, as observed also in [GM95, (3.1)]. By the proof of [GM95, Lemma 2.3], we get thatB X ε contains an isometric copy of ∆ n+1 k (ǫ) with totally geodesic interior and vertexes given by j=1 . In the assumption of an upper bound on the curvature, there is not a general natural notion of dimension. However, one can still introduce a geometric dimension GeomDim(X) of a CBA space X, which is defined as the smallest function defined on CBA spaces such that a) GeomDim(X) = 0 whenever X is a discrete space, and b) GeomDim(X) ≥ 1 + GeomDim(Σ p X) for every p ∈ X, Σ p X being the space of directions; see [Kle99]. B. Kleiner proved in particular that GeomDim(X) is equal to the topological dimension of the space. Moreover, whenever GeomDim(X) < +∞, it holds that GeomDim(X) is the greater integer q such that there is an isometric embedding of the standard unit sphere S n−1 ⊂ R n into Σ p X for some p ∈ X. By the definition of the space of directions and by the monotonicity condition for upper curvature bounds on metric space (see [BBI01,Section 4.3.1]), one easily gets Proposition 19. Let (X, d) be a CBA(k) Alexandrov space. Suppose that there is an isometric embedding of the standard unit sphere S n−1 ⊂ R n into Σ x X for some x ∈ X. Then Scal

Curvature measure of CBB(k) surfaces
In this section we focus on surfaces with lower bounded curvature. Let (S, d) be a 2-dimensional topological surface, whose metric is CBB(k) for some k ∈ R. (S, d) is in particular a surface of bounded integral curvature (BIC) so that it supports a well-defined curvature measure ω, see for instance [Res93], [AZ67] or the survey [Tro09]. This latter measure constitutes a generalization of the Riemannian curvature of a surface to the non-smooth setting, i.e. dω = Sect g dA g = 1 2 Scal g dA g whenever the distance d of (S, d) is induced by a smooth Riemannian metric g on S. In fact ω can be defined as the weak limit (in the sense of measures) of the sectional curvature of an approximating sequence of smooth metrics. In [Mac98],

Towards scalar curvature measure
Kapovitch, Lytchak and Petrunin recently proposed a notion of metric scalar curvature ν on a metric measured space (X, d, µ), called metric-measure curvature, µ being a positive Radon measure on (X, d). For x ∈ X define b r (x) := µ(B r (x)) and introduce the deviation measure v r on X, absolutely continuous with respect to µ, given by µ.
This is primarly motivated by the fact that, when X = (M, g) is a Riemannian manifold endowed with Riemannian distance d = d g and volume µ = Vol g , according to (1) the one-parameter family {v r /r 2 } r>0 converges as a measure to 1 6(n+2) Scal g d Vol g as r → 0. Accordingly, one expects the limit measure ν := lim r→0 v r /r 2 , whenever it exists, to be a good candidate to replace the scalar curvature on measure metric spaces. In [KLP17] it is in particular proven that the family of measure v r /r 2 is uniformly bounded on surfaces with bounded integral surfaces, so that it exists a (a priori non unique) metric-measure curvature ν. Note that, unfortunately, ν do not coincide in general with the natural intrinsic curvature measure of the surface; see [KLP17, Exemple 1.14].
In this section we adapt the construction of [KLP17] to define an extent-curvature measure and a boundary-extent-curvature-measure which generalize the Scalar curvature (measure) on a Riemannian manifold using (3) or (4) instead of (1). As it is the case for the mm-curvature measure, we will see that extent-curvature measures and boundary-extent-curvature measures exist on BIC surfaces.
In the following, let (S, d) be a surface of bounded integral curvature, and consider its area measure coinciding with the two-dimensional Hausdorff measure H 2 . For any fixed scale r, consider the deviation measure e r on X, absolutely continuous with respect to H 2 , given by We will say that (S, d) has locally finite extent curvature if the family of measures {e r /r 2 } 0<r≤1 is uniformly bounded. Whenever a limit lim r→0 e r /r 2 exists, it is called extent curvature measure. Similarly, we can define and say that (S, d) has locally finite boundary extent curvature if the family of measures {∂e r /r 2 } 0<r≤1 is uniformly bounded. Whenever a limit lim r→0 ∂e r /r 2 exists, it is called boundary extent curvature measure.
We have the following result.
Theorem 21. Let (S, d) be a surface with bounded integral curvature.
Proof. The proof is essentially the same as the proof of [KLP17, Theorem 4.2]. We will sketch the relevant changes for completeness.
The main ingredient is the analogue of [KLP17, Lemma 4.1] Lemma 22. There exists some δ 0 > 0 with the following property. Let S be a surface with bounded integral curvature and let ω be its curvature measure. Let x ∈ X be a point and let r > 0 be such thatB d r (x) is compact, |ω|(B d r (x)) < δ 0 and B d r (x) ⊂ U when U is an open subset of S homeomorphic to R 2 . Then and Proof. To prove (43), it suffices to prove that for all 0 < s < r Moreover it is enough to check this latter inequality on polyhedral metrics homeomorphic to R 2 . In fact, let s < s ′ < t. According to [Res93,Theorem 4.3] we take a polyhedral sequence {d k } of metrics on S approaching d uniformly and tamely, see Theorm 8.4.3 and the subsequent discussion [Res93], or [AZ67, Theorem 7 of Chapter VII]. Without loss of generality let sup y,y ′ ∈U |d(y, y ′ ) − d k (y, y ′ )| < 1 k .
Suppose that for any polyhedral metric d k . Let x j,k , j = 1, 2, 3 be a 3-extender ofB d k s (x). Then x j,k ∈ B d s+ 1 k (x) and xt 3 (B d s (x)) ≥ lim sup k→∞ xt 3 (B d k s (x)). Similarly, if {y j } j=1,2,3 is a 3-extender ofB d s (x), then y j ∈ B d k s+1/k (x) so that xt 3 (B d s (x)) ≤ lim inf k→∞ xt 3 (B d k s+1/k (x)). Then the proof is the same as in [KLP17], and it exploits the construction of an explicit completion of the local polyhedral surface to a global polyhedral BIC metric on R 2 with controlled curvature, as well as the biLipschitz maps to R 2 provided by [BL03].
To conclude, one has just to remark that on regions (1 + δ ′ )-biLipschitz diffeomorphic to Euclidean open sets it holds for r small enough (compare with [KLP17, Section 2.4]).
The proof of (44) is essentially the same. One has just to keep in account that by definition |∂xt 3 (B r (x)) − xt 3 (B r (x))| < η whenever the three points realizing xt 3 (B r (x)) are in B r (x) \ B r−η/6 (x).
We prove (1), the proof of (2) being the same. The result is local so that it suffices to prove it in a neighborhood of an arbitrary point x 0 ∈ S. Let X be a neighborhood of x 0 homeomorphic to R 2 and satisfying |ω|(X \ {x 0 }) < min{δ 0 ; 1/3}, δ 0 being the positive constant in Lemma 22. Let A ⊂ X compact. We want to prove that there exist an ε > 0 and a constant C = C(ǫ) > 0 such that for all 0 < r < ǫ one has (45) |e r |(A) ≤ Cr 2 First note that for any x ∈ S and positive s, it holds xt 3 (B s (x)) ≤ 2s and According to [KLP17], up to take a smaller ǫ we have that for all 0 < 3r < ǫ H 2 (B 3r (x 0 )) < 9r 2 4ǫ .