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Let $H_{f}$ be the convex hull of the graph of $f$, the concave function on the top of $H_{f}$ is denoted by $ { { \\varphi } }_{1,f}( { { \\mathbf x } })=\\max \\{y:( { { \\mathbf x } },y)\\in H_{f} \\}$ and $ { { \\varphi } }_{2,f}( { { \\mathbf x } })=\\min \\{y:( { { \\mathbf x } },y)\\in H_{f} \\}$ denotes the c"},"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":"1603.09162","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-03-30T12:59:29Z","cross_cats_sorted":[],"title_canon_sha256":"2b6db5caa5c8fb943aaa3581a2323c4720f271b71b1c1008aedc8c5e428a7bf5","abstract_canon_sha256":"41a6d76cf0dcc7a8d7a364451d301e732d9045d23e40bf366021588541973be7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:24.097440Z","signature_b64":"TEhdUR8GVzmdBos5as5FpImtfjZWNyULLzO7cEPE7jbNnwvowCLmBpUUQeHuXtAQ9eW5vsT6j9GaBhW0d76AAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92a124ac7ca5d87f78645981eea6aa0e3a2929103aca25d306301fe4c318c725","last_reissued_at":"2026-05-18T01:16:24.096930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:24.096930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multifractal properties of convex hulls of typical continuous functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Zoltan Buczolich","submitted_at":"2016-03-30T12:59:29Z","abstract_excerpt":"We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on $[0,1]^{d}$. 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Let $H_{f}$ be the convex hull of the graph of $f$, the concave function on the top of $H_{f}$ is denoted by $ { { \\varphi } }_{1,f}( { { \\mathbf x } })=\\max \\{y:( { { \\mathbf x } },y)\\in H_{f} \\}$ and $ { { \\varphi } }_{2,f}( { { \\mathbf x } })=\\min \\{y:( { { \\mathbf x } },y)\\in H_{f} \\}$ denotes the c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09162","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.09162","created_at":"2026-05-18T01:16:24.096998+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.09162v2","created_at":"2026-05-18T01:16:24.096998+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.09162","created_at":"2026-05-18T01:16:24.096998+00:00"},{"alias_kind":"pith_short_12","alias_value":"SKQSJLD4UXMH","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"SKQSJLD4UXMH66DE","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"SKQSJLD4","created_at":"2026-05-18T12:30:44.179134+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY","json":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY.json","graph_json":"https://pith.science/api/pith-number/SKQSJLD4UXMH66DELGA65JVKBY/graph.json","events_json":"https://pith.science/api/pith-number/SKQSJLD4UXMH66DELGA65JVKBY/events.json","paper":"https://pith.science/paper/SKQSJLD4"},"agent_actions":{"view_html":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY","download_json":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY.json","view_paper":"https://pith.science/paper/SKQSJLD4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.09162&json=true","fetch_graph":"https://pith.science/api/pith-number/SKQSJLD4UXMH66DELGA65JVKBY/graph.json","fetch_events":"https://pith.science/api/pith-number/SKQSJLD4UXMH66DELGA65JVKBY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY/action/storage_attestation","attest_author":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY/action/author_attestation","sign_citation":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY/action/citation_signature","submit_replication":"https://pith.science/pith/SKQSJLD4UXMH66DELGA65JVKBY/action/replication_record"}},"created_at":"2026-05-18T01:16:24.096998+00:00","updated_at":"2026-05-18T01:16:24.096998+00:00"}