{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:WTF4A4S7GQH63RB7CVMSDD45CM","short_pith_number":"pith:WTF4A4S7","canonical_record":{"source":{"id":"2601.20117","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-01-27T23:17:59Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"e33637c57d08c1d369a95984eefafe83941a04c5a3a400a542f5b83f1ab7a577","abstract_canon_sha256":"06de80315019834ae5b631644440ac8ee19aa185720ab3a4e0ba6ebc877bda5d"},"schema_version":"1.0"},"canonical_sha256":"b4cbc0725f340fedc43f1559218f9d131f0a8a9e80b30163949e0a9630b48297","source":{"kind":"arxiv","id":"2601.20117","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.20117","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"arxiv_version","alias_value":"2601.20117v2","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.20117","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"pith_short_12","alias_value":"WTF4A4S7GQH6","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"pith_short_16","alias_value":"WTF4A4S7GQH63RB7","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"pith_short_8","alias_value":"WTF4A4S7","created_at":"2026-07-02T00:18:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:WTF4A4S7GQH63RB7CVMSDD45CM","target":"record","payload":{"canonical_record":{"source":{"id":"2601.20117","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-01-27T23:17:59Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"e33637c57d08c1d369a95984eefafe83941a04c5a3a400a542f5b83f1ab7a577","abstract_canon_sha256":"06de80315019834ae5b631644440ac8ee19aa185720ab3a4e0ba6ebc877bda5d"},"schema_version":"1.0"},"canonical_sha256":"b4cbc0725f340fedc43f1559218f9d131f0a8a9e80b30163949e0a9630b48297","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T00:18:26.928565Z","signature_b64":"0fyQaWTlJqyo0Y9vQUQ8BwW/mHa7gqS/4UPibA40d1CD4J5UE8a5C/FzmbLpv87EdM6eW+qC2oFwa0FofTYGDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4cbc0725f340fedc43f1559218f9d131f0a8a9e80b30163949e0a9630b48297","last_reissued_at":"2026-07-02T00:18:26.927780Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T00:18:26.927780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2601.20117","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-02T00:18:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1GQpeDN9aW8WaJIrrG+zjTmLK4uEmIYEZvYYVfonoi3yGTJOcZcsC8t0rVgsETgPl2rv/NfQd9ihwG5oGds8BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T19:12:59.814301Z"},"content_sha256":"ffb6b0a2ab72f0d08f3f2a5186053c1b34948c2c4bd04ac8a154ec39cf524339","schema_version":"1.0","event_id":"sha256:ffb6b0a2ab72f0d08f3f2a5186053c1b34948c2c4bd04ac8a154ec39cf524339"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:WTF4A4S7GQH63RB7CVMSDD45CM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Fourier Mean Bodies of a Convex Body","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Artem Zvavitch, Auttawich Manui, Dylan Langharst","submitted_at":"2026-01-27T23:17:59Z","abstract_excerpt":"In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_pK$ of a convex body $K\\subset\\mathbb R^n$, $p>-1$, which have since become important objects in geometric tomography. In this paper we study the Fourier transforms of the radial functions of $R_pK$. This leads to a new family of star-shaped sets $F_pK$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_pK$ and $F_pK$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_pK$; this includes affine invariance"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.20117","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.20117/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-02T00:18:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7/ju2WMTMRibHXCp3+2UpICoLdJVoj5ielXkvu8kHpa4T1wNUlpQUFxG+GV3v0ts186n7PpzR4r0HLrpvEa1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T19:12:59.814709Z"},"content_sha256":"ecd1993e85c47e475b03dda756bd0dbce09b1157b9fc3fe96bb33603343ac297","schema_version":"1.0","event_id":"sha256:ecd1993e85c47e475b03dda756bd0dbce09b1157b9fc3fe96bb33603343ac297"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WTF4A4S7GQH63RB7CVMSDD45CM/bundle.json","state_url":"https://pith.science/pith/WTF4A4S7GQH63RB7CVMSDD45CM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WTF4A4S7GQH63RB7CVMSDD45CM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T19:12:59Z","links":{"resolver":"https://pith.science/pith/WTF4A4S7GQH63RB7CVMSDD45CM","bundle":"https://pith.science/pith/WTF4A4S7GQH63RB7CVMSDD45CM/bundle.json","state":"https://pith.science/pith/WTF4A4S7GQH63RB7CVMSDD45CM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WTF4A4S7GQH63RB7CVMSDD45CM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:WTF4A4S7GQH63RB7CVMSDD45CM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"06de80315019834ae5b631644440ac8ee19aa185720ab3a4e0ba6ebc877bda5d","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-01-27T23:17:59Z","title_canon_sha256":"e33637c57d08c1d369a95984eefafe83941a04c5a3a400a542f5b83f1ab7a577"},"schema_version":"1.0","source":{"id":"2601.20117","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.20117","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"arxiv_version","alias_value":"2601.20117v2","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.20117","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"pith_short_12","alias_value":"WTF4A4S7GQH6","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"pith_short_16","alias_value":"WTF4A4S7GQH63RB7","created_at":"2026-07-02T00:18:26Z"},{"alias_kind":"pith_short_8","alias_value":"WTF4A4S7","created_at":"2026-07-02T00:18:26Z"}],"graph_snapshots":[{"event_id":"sha256:ecd1993e85c47e475b03dda756bd0dbce09b1157b9fc3fe96bb33603343ac297","target":"graph","created_at":"2026-07-02T00:18:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2601.20117/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_pK$ of a convex body $K\\subset\\mathbb R^n$, $p>-1$, which have since become important objects in geometric tomography. In this paper we study the Fourier transforms of the radial functions of $R_pK$. This leads to a new family of star-shaped sets $F_pK$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_pK$ and $F_pK$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_pK$; this includes affine invariance","authors_text":"Artem Zvavitch, Auttawich Manui, Dylan Langharst","cross_cats":["math.FA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-01-27T23:17:59Z","title":"On the Fourier Mean Bodies of a Convex Body"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.20117","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ffb6b0a2ab72f0d08f3f2a5186053c1b34948c2c4bd04ac8a154ec39cf524339","target":"record","created_at":"2026-07-02T00:18:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"06de80315019834ae5b631644440ac8ee19aa185720ab3a4e0ba6ebc877bda5d","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-01-27T23:17:59Z","title_canon_sha256":"e33637c57d08c1d369a95984eefafe83941a04c5a3a400a542f5b83f1ab7a577"},"schema_version":"1.0","source":{"id":"2601.20117","kind":"arxiv","version":2}},"canonical_sha256":"b4cbc0725f340fedc43f1559218f9d131f0a8a9e80b30163949e0a9630b48297","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b4cbc0725f340fedc43f1559218f9d131f0a8a9e80b30163949e0a9630b48297","first_computed_at":"2026-07-02T00:18:26.927780Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-02T00:18:26.927780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0fyQaWTlJqyo0Y9vQUQ8BwW/mHa7gqS/4UPibA40d1CD4J5UE8a5C/FzmbLpv87EdM6eW+qC2oFwa0FofTYGDw==","signature_status":"signed_v1","signed_at":"2026-07-02T00:18:26.928565Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.20117","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ffb6b0a2ab72f0d08f3f2a5186053c1b34948c2c4bd04ac8a154ec39cf524339","sha256:ecd1993e85c47e475b03dda756bd0dbce09b1157b9fc3fe96bb33603343ac297"],"state_sha256":"b37020f12fca75405468d3ba994a9087a81cbb52bd40e4355e08f98431871805"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BJvkR3DRJMV1nEIRypBhJockisSGrBzMmJ9A7OyXheEOTnh4daBhW0fiW2lj3xe2KNa35r64agR5doyNmmlGDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T19:12:59.816918Z","bundle_sha256":"6a0994b4823b7baacf2dc8004aa19f4e767f98e198aac1ac089de9bc645ad096"}}