| Dual norm identity in the tangent cone | statement | lemma 4.13 | arxiv |
| $\kappa$-comparison angle | definition | definition 2.5 | arxiv |
| 2C79QP3EBA | statement | proposition 2.4 | arxiv |
| 2QMZWWCCWB | definition | definition | arxiv |
| 2TEX5390VK | statement | proposition 4.5 | arxiv |
| Quadratic variance inequality | statement | proposition 4.6 | arxiv |
| First variation formula | statement | proposition 2.13 | arxiv |
| Fr\'echet-cosine inequality | statement | lemma 4.2 | arxiv |
| Descent in expectation | statement | lemma 5.7 | arxiv |
| 973DMBJ594 | statement | theorem 5.21 | arxiv |
| Yokota's second localization | statement | lemma B.4 | arxiv |
| Polyak--\Lojasiewicz inequality | statement | theorem 4.11 | arxiv |
| High-probability bound for iterated barycenters | statement | corollary 5.12 | arxiv |
| AG0Y0J5RAP | statement | theorem 5.15 | arxiv |
| Quantitative four-point inequality | statement | proposition 3.3 | arxiv |
| BGEXQ40ZM6 | statement | theorem 5.22 | arxiv |
| BGNANMWWVT | statement | corollary B.2 | arxiv |
| Stability of iterated barycenters | statement | theorem 5.5 | arxiv |
| Alexandrov angle | definition | definition 2.7 | arxiv |
| CDYGTMGWBJ | statement | lemma 3.7 | arxiv |
| Expected error for empirical barycenters | statement | theorem 5.1 | arxiv |
| CPH8EZJJP8 | statement | corollary 5.25 | arxiv |
| Metric version of Carathéodory's theorem | statement | theorem 5.24 | arxiv |
| Ekeland's variational principle | statement | lemma B.1 | arxiv |
| Stability with a common initial point | statement | proposition 3.4 | arxiv |
| Carathéodory's theorem | statement | theorem 5.23 | arxiv |
| Iterated barycenter | definition | definition 5.4 | arxiv |
| Tangent cone and log function | definition | definition 2.11 | arxiv |
| Euclidean cone over a metric space | definition | definition 3.2 | arxiv |
| Expected error for iterated barycenters | statement | theorem 5.9 | arxiv |
| Absolute continuity and metric speed | definition | definition 2.1 | arxiv |
| Stability with common terminal point | statement | proposition 3.6 | arxiv |
| Barycentric hull | definition | definition 5.14 | arxiv |
| Reshetnyak's four point inequality | statement | proposition 3.1 | arxiv |
| HCZS013QEZ | statement | corollary B.5 | arxiv |
| Convex hull representation | statement | lemma 5.18 | arxiv |
| Global curvature upper bounds and $\CAT(\kappa)$ spaces | definition | definition 2.2 | arxiv |
| Geodesics in the Euclidean cone | statement | proposition A.1 | arxiv |
| $\kappa$-domain | definition | definition 2.3 | arxiv |
| Angle comparison | statement | proposition 2.8 | arxiv |
| Projection onto a geodesically convex set | statement | proposition 2.14 | arxiv |
| Directional Fr\'echet-cosine inequality | statement | theorem 4.3 | arxiv |
| Wasserstein-1 Lipschitz stability | statement | proposition 4.8 | arxiv |
| SFNHTYXM7J | statement | theorem 5.16 | arxiv |
| Discrete recurrence estimates | statement | lemma 5.8 | arxiv |
| Distance to the barycenter via tangent directions | statement | proposition 4.15 | arxiv |
| Descending metric slope | definition | definition 4.10 | arxiv |
| High probability bounds for empirical barycenters | statement | corollary 5.3 | arxiv |
| Geometry of tangent cones | statement | proposition 2.12 | arxiv |
| Cosine inequality along geodesics | statement | proposition 2.10 | arxiv |
| Directional derivative of the Fr\'echet function | statement | proposition 4.1 | arxiv |
| Equality of barycentric and convex hulls on model spaces | statement | theorem 5.27 | arxiv |
| Weighted distance | definition | definition 3.5 | arxiv |
| X5PBAKE3NQ | statement | lemma 5.20 | arxiv |
| Sparse approximations of barycentric hulls | statement | theorem 5.26 | arxiv |
| Existence and uniqueness {(see | statement | theorem 4.4 | arxiv |
| Yokota's first localization | statement | lemma B.3 | arxiv |
| Cosine comparison in a $\kappa$-domain | statement | proposition 2.9 | arxiv |
| Z7862QC0SY | statement | lemma 5.19 | arxiv |
