A note on flatness of non separable tangent cone

Given a probability measure P on an Alexandrov space S with curvature bounded below, we prove that the support of the pushforward of P on the tangent cone at its (exponential) barycenter is a subset of a Hilbert space, without separability of the tangent cone.


Introduction
Barycenter of a probability measure P (a.k.a.Fréchet means) provides an extension of expectation on Euclidean space to arbitrary metric spaces.We present here a useful tool for the study of barycenters on Alexandrov spaces with curvature bounded below: the support of log b ⋆ #P in the tangent cone at the barycenter is included in a Hilbert space.This result has been stated in [Yok12] as Theorem 45, however the proof is not written.Moreover, there is an extra assumption of support of log b ⋆ #P being separable, which does not even seem to be a consequence of the support of P being separable.This paper present a proof of this result, without this extra separable assumption.The proof is essentially the one of Theorem 45 of [Yok12], with needed approximations dealt with a bit differently.

Setting and main result
We use a classical notion of curvature bounded below for geodesic spaces, referred to as Alexandrov curvature.We recall several notions whose formal definitions can be found for instance in [BBI01] or in the work in progress [AKP19].
For a metric space (S, d), we denote by P2(S) that set of probability measures P on S with finite moment of order 2 (i.e.there exists x ∈ S such that d 2 (x, y)dP(y) < ∞).The support of a measure P will be denoted by supp P.
A geodesic space is a metric space (S, d) such that every two points x, y ∈ S at distance is connected by a curve of length d(x, y).
Then, the tangent cone TpS is defined as the completion of TpS.We will use the notation for u, v ∈ TpS, We will often identify a point γ(t) ∈ S with (γ, t) ∈ TpS.Although such γ might not be unique, we will implicitly assume the choice of a measurable map log p : S → TpS, called logarithmic map, such that for all x ∈ S, there exists a geodesic γ emanating from p such that, for some t > 0, γ(t) = x and log p (x) = (γ, t).
Then the pushforward of P by log p will be denoted by P# log p .The tangent cone is not necessarily a geodesic space (see [Hal00]), however, it is included in a geodesic space -namely the ultratangent space (see for instance Theorem 14.4.2 and 14.4.1 of [AKP19]) that is an Alexandrov space with curvature bounded below by 0.
The tangent cone TpS contains the subspace Linp of all points with an opposite, formally defined as follows.A point u belongs to Linp ⊂ TpS if and only if there exists v ∈ TpS such that u p = v p and u, v p = − u 2 p .
Our main result is based on the following Theorem.
Theorem (Theorem 14.5.4 in [AKP19]).The set Linp equipped with the induced metric of TpS is a Hilbert space.
A point b ⋆ ∈ S is a barycenter of the probability measure Such barycenter might not be unique, neither exist.However, when they exist, they satisfy x, y b ⋆ dP ⊗ P(x, y) = 0. (1) A point b ⋆ ∈ S satisfying (1) is called an exponential barycenter of P.
We can now state our main result.This result allows to prove the following Corollary, that has been implicitly used in [ALP18].

Proofs
Recall that we always identify a point in S and its image in the tangent cone TpS by the log p map.
Proof of Corollary 2. We check that x → x, b b ⋆ is a convex and concave function in Lin b ⋆ S. Let t ∈ (0, 1), x0, x1 in Lin b ⋆ S, and set xt = (1 − t)x0 + tx1.Since the tangent cone is included in an Alexandrov space with curvature bounded below by 0 on one hand, and Lin b ⋆ is a Hilbert space on the other hand, The same lines applied to −x0 and −x1 gives the converse inequality The second statement follows from the fact that b ⋆ is a Pettis integral of the pushforward of P onto Lin b ⋆ ⊂ T b ⋆ S, as a direct consequence of Theorem 1.
Proof of Theorem 1.Let x ∈ supp P. For U = {x}, use Lemma 5 with Then, since x, y b ⋆ dP(y) = 0 by Lemma 3, letting δ → 0, one gets Thus, lim is a subspace of an Alexandrov space of curvature bounded below by 0, we also have as n, k → ∞.Thus (ȳ n )n correspond to a Cauchy sequence in the space of direction, and thus admits a limit in T b ⋆ S -since its "norm" also admits a limit d(b ⋆ , x).Finally, its limit ȳ satisfies cos ∠(↑ x b ⋆ , ↑ ȳ b ⋆ ) = −1, and therefore, it is the opposite ȳ = −x.
Lemma 3 (Proposition 1.7 of [Stu99] for non separable metric space).Suppose (S, d) is an Alexandrov space with curvature bounded below.Then, for any probability measure Q ∈ P2(S), Proof.For brevity, we will adopt the notation Qg for gdQ.
The result for Q finitely supported is the Lang-Schroeder inequality (Proposition 3.2 in [LS97]).Thus, we just need to approximate Q ⊗ Q ., .b ⋆ by some Qn ⊗ Qn , ., b ⋆ for some finitely supported Qn.
To do this, for i.i.d.random variable (Xi)i of common law Q, denote Qn the empirical measure.Since S is not separable, we can not apply the fundamental theorem of statistics that ensures almost sure weak convergence of Qn to Q.However, for a measurable function f : S × S → R, such that Q ⊗ Qf 2 < ∞, we get the following bound And thus, Qn ⊗ Qnf → Q ⊗ Qf in L 2 (Q ⊗∞ ) and so there exists a (deterministic) probability measure Qn supported on n points, such that Qn ⊗Qnf → Q⊗Qf .We thus proved the first result applying f = ., .b ⋆ .Now applying this first result to the measure Qε : (3)

Theorem 1 .
Let (S, d) be an Alexandrov space with curvature bounded below by some κ ∈ R and P ∈ P2(S).If b ⋆ ∈ S is an exponential barycenter of P, then the supp(log b ⋆ #P) ⊂ Lin b ⋆ S. In particular, supp(log b ⋆ #P) is included in a Hilbert space.
Such shortest curves are called geodesics.For κ ∈ R, the model space (Mκ, dκ) denotes the 2-dimensional surface of constant Gauss curvature κ.
Using angles, we can define the tangent cone TpS at p ∈ S as follows.First define T ′ p S as the (quotient) set Γx × R + , equipped with the (pseudo-)metric defined by (γ1, t) − (γ2, s) 2 p