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This bound is sharp for each $n\\geq 3$.\n  In a companion paper, we have asymptotically determined this maximum for points in $R^"},"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":"1404.2624","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-09T20:50:41Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"b0c6dd6a4564df767d8bad54a415b3af17c1424a71c5ec73c0f39b6f410170f8","abstract_canon_sha256":"443f5baddf8cf8d016dfcd3d385096077eaf68bfa0dc8223a40e22d04f1da047"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:58.764828Z","signature_b64":"WDEGZ4/wBSvXCyR2OqYPE87bYv6QziqBKwdUJsEQEqB3B+EM2kPLGxqGZz4Uq57PfOXsQIjaIFc321YrQT9CCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5a59e138daa71e4fa41ab5691a3aa7978e936cf59d0e0d3ee4e6a16467870e8","last_reissued_at":"2026-05-18T01:33:58.764224Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:58.764224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Double-normal pairs in the plane and on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"J\\'anos Pach, Konrad J. 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