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When U \\subset V \\subset H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let D=diam_V (U) \\in [0,+\\infty] be the diameter of the smaller subsets with respect to the large. Then for every x,y in U we have  d_V(x,y) \\leq tanh (D/4) d_U(x,y).\n  In dimension one, this contraction principle was established by Birkhoff for the Hilbert metric of fi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1102.4212","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-21T13:07:01Z","cross_cats_sorted":["math.CV","math.DG"],"title_canon_sha256":"ed918e678fa16b12c09054445a0ac2f145a1283fe2acf06681e14970d4d8d9d1","abstract_canon_sha256":"d844c1e9b402fe70935a5f227d23c5b4b5da7d9f79fc7279b3d52d65077b6138"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:10.724083Z","signature_b64":"X0PTlkfIeELUJ+Kw/bDrFyp7xBSBkRXFaZQrqrtf7Tb0FU7ownQXdQ9VablvbyJbB6A4KSiNoDnl/ELEWF32Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4648668030e6a9b823eee8d5ab6ed086d55ab64426ee85ac2e1c13014a3b4a97","last_reissued_at":"2026-05-18T04:28:10.723338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:10.723338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A uniform contraction principle for bounded Apollonian embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.DG"],"primary_cat":"math.MG","authors_text":"Hans Henrik Rugh, Lo\\\"ic Dubois","submitted_at":"2011-02-21T13:07:01Z","abstract_excerpt":"Let H denote the standard one-point completion of a real Hilbert space. 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