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Our starting point is the first variation formula of Falah and Rotem (Calc. Var. and PDE, 2026), which associates to each log-concave function a pair of surface area measures. Using the additivity of these measures, we define a canonical Blaschke sum and Blaschke homothety on the class of log-concave functions, uniquely determined up to translation. We establish the basic algebraic properties of these operations, define the associated Blaschke symmetral, and show that","authors_text":"Effrosyni Chasioti, Steven Hoehner","cross_cats":["math.MG"],"headline":"Blaschke addition on log-concave functions produces affine isoperimetric inequalities with radial maximizers.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-05-17T23:01:40Z","title":"Blaschke operations on log-concave functions and affine isoperimetric inequalities"},"references":{"count":31,"internal_anchors":0,"resolved_work":31,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Applebaum.Probability on Compact Lie Groups, volume 70 ofProbability Theory and Stochastic Mod- elling","work_id":"d037a2d9-0181-44b6-880e-b5021c4673d3","year":2014},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"G. Bianchi, R. J. Gardner, and P. Gronchi. 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Functional versions ofLp-affine surface area and entropy inequalities.International Mathematics Research Notices, 4:1223–12","work_id":"1171a442-5932-4008-9c9e-17fa6ec9b810","year":2016}],"snapshot_sha256":"4bb2d94bfb33b752ebe97bb2dc0bfa3bd36355dff68b72d6ebac879efd733578"},"source":{"id":"2605.17688","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T21:51:05.952773Z","id":"54453162-0da9-47f2-9a50-74cf323bf6b8","model_set":{"reader":"grok-4.3"},"one_line_summary":"Defines canonical Blaschke operations on log-concave functions and derives associated affine isoperimetric inequalities plus Kneser-Süss-type results.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Blaschke addition on log-concave functions produces affine isoperimetric inequalities with radial maximizers.","strongest_claim":"We prove affine isoperimetric inequalities for log-concave functions; in particular, the functional affine surface area is maximized, under fixed first quermassintegral, by radially symmetric functions, and satisfies a Blaschke-concavity property.","weakest_assumption":"The surface area measures arising from the first variation formula of Falah and Rotem are additive, which is used to define the canonical Blaschke sum uniquely up to translation (abstract, paragraph 2)."}},"verdict_id":"54453162-0da9-47f2-9a50-74cf323bf6b8"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:561460c712f34e3ad9eaf5e3471984be0b425f25c96a44f17cd6a38237e9ea8a","target":"record","created_at":"2026-05-20T00:04:52Z","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":"02b6bc7c6900ec6ae621a5ea23e760c444af593603b80e707066eafb3e1360ac","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-05-17T23:01:40Z","title_canon_sha256":"4608b235a496bee67eac0e47066a823d68de1702c5390cdea5e71017a375ca14"},"schema_version":"1.0","source":{"id":"2605.17688","kind":"arxiv","version":1}},"canonical_sha256":"1010f1e937cb592d22f8e04054050eea9fc2e06897d98b19c885fb9696f67bbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1010f1e937cb592d22f8e04054050eea9fc2e06897d98b19c885fb9696f67bbd","first_computed_at":"2026-05-20T00:04:52.860620Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:52.860620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2cudOMSQr1MNH7aWNqLUsuFlk7atQ0zmyMcvhOtvsQYgypntDpaRt+XDsuppNZbW2Zhh/zkXAneTAp7JJSxRAw==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:52.861387Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17688","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:561460c712f34e3ad9eaf5e3471984be0b425f25c96a44f17cd6a38237e9ea8a","sha256:ce30fdf32fbc9459668b5c105171052e71b350813ac18aa684bf03fe32faf5ac"],"state_sha256":"f942a897fd4bc7a5e6be190f0224f8ccf4830b9db9484d5f3f32c496045654f9"}