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We define the Diederich-Forn{\\ae}ss index with respect to a family of functions to be the supremum over the set of all exponents $0<\\eta<1$ such that there exists a function $\\rho_\\eta$ in this family such that $-\\rho_\\eta$ is comparable to the distance to the boundary of $\\Omega$ on $\\Omega$ and such that $-(-\\rho_\\eta)^\\eta$ is plurisubharmonic on $\\Omega$. We first prove that computing the Diederich-Forn{\\ae}ss index with respect to the family of upper semi-continuous functions is the same as computing the Diederich-Forn{\\ae}"},"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":"1907.03689","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2019-07-08T15:52:20Z","cross_cats_sorted":[],"title_canon_sha256":"92e1be6a06670ccbe3304e34d8e679724fe149fffdfdb264a44306bd1e5ff81c","abstract_canon_sha256":"17a2f5888268ae30b3e4f29908cafc76a07f068eb2a8030155a2b98d699d5106"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:14.022711Z","signature_b64":"OQ11nRtCa2TpcBbwsINKa4bnDSC2P1ItBAkzJmN0dI49E0ObYQ+CRPCqwnlMqXQbzk+YpCAHA2oWXfqMYuenBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"171a16f9995c204eda9419c0061be434990d26c9f810b71f8b98b88a98a46e3f","last_reissued_at":"2026-05-17T23:41:14.022296Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:14.022296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Competing Definitions for the Diederich-Forn{\\ae}ss Index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Phillip S. 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