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Does the radial projection of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that these are the only obstructions: if $\\dim_{\\mathrm{H}} E > 0$, and $E$ does not lie on a line, then there exists a point in $x \\in E$ such that the radial projectio"},"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":"1710.11053","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-30T16:32:16Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"035b116554cd53f67f3edaca078875f2aa5d960fa2815bc373eae846b3b2e0e9","abstract_canon_sha256":"229187b4c3fba8a517a33106e092682c90986e356e819c3c95c975622dbb7da7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:05.353697Z","signature_b64":"3DcwUr/lN6inpbbyeUIFs4NMCPOD9hbndIHEv+SU8XgT3CbDCOfeLXCoEJRzFZ5FV2jk/C5HVkO6IYhJD7scCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b118525e5e54d3a79c3da51eb3608df274553a27eab8ac17f2f9f3dcfdfd1ac0","last_reissued_at":"2026-05-17T23:58:05.353193Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:05.353193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the dimension and smoothness of radial projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2017-10-30T16:32:16Z","abstract_excerpt":"This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces.\n  To introduce the first one, assume that $E,K \\subset \\mathbb{R}^{2}$ are non-empty Borel sets with $\\dim_{\\mathrm{H}} K > 0$. Does the radial projection of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that these are the only obstructions: if $\\dim_{\\mathrm{H}} E > 0$, and $E$ does not lie on a line, then there exists a point in $x \\in E$ such that the radial projectio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11053","kind":"arxiv","version":3},"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":"1710.11053","created_at":"2026-05-17T23:58:05.353276+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.11053v3","created_at":"2026-05-17T23:58:05.353276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.11053","created_at":"2026-05-17T23:58:05.353276+00:00"},{"alias_kind":"pith_short_12","alias_value":"WEMFEXS6KTJ2","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WEMFEXS6KTJ2PHB5","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WEMFEXS6","created_at":"2026-05-18T12:31:53.515858+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/WEMFEXS6KTJ2PHB5UUPLGYEN6J","json":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J.json","graph_json":"https://pith.science/api/pith-number/WEMFEXS6KTJ2PHB5UUPLGYEN6J/graph.json","events_json":"https://pith.science/api/pith-number/WEMFEXS6KTJ2PHB5UUPLGYEN6J/events.json","paper":"https://pith.science/paper/WEMFEXS6"},"agent_actions":{"view_html":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J","download_json":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J.json","view_paper":"https://pith.science/paper/WEMFEXS6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.11053&json=true","fetch_graph":"https://pith.science/api/pith-number/WEMFEXS6KTJ2PHB5UUPLGYEN6J/graph.json","fetch_events":"https://pith.science/api/pith-number/WEMFEXS6KTJ2PHB5UUPLGYEN6J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J/action/storage_attestation","attest_author":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J/action/author_attestation","sign_citation":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J/action/citation_signature","submit_replication":"https://pith.science/pith/WEMFEXS6KTJ2PHB5UUPLGYEN6J/action/replication_record"}},"created_at":"2026-05-17T23:58:05.353276+00:00","updated_at":"2026-05-17T23:58:05.353276+00:00"}