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We show that if $B$ is not a cube then $B$ can be illuminated by strictly less than $2^n$ sources of light. This confirms the Hadwiger--Gohberg--Markus illumination conjecture for unit balls of $1$-symmetric norms in $R^n$ for al"},"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":"1606.08976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-06-29T06:47:18Z","cross_cats_sorted":[],"title_canon_sha256":"63ee3e55a54bbc99647f8b6e2a5459df5084bbb0637fbd1fd32f5d9cba850e8d","abstract_canon_sha256":"27370a9dd6bc2e77a82192414a3e139d5fee441d883a951ef48eca119894fb8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:13.097454Z","signature_b64":"srsh++uqAw/MP5xGhVwakNTGMohY91sinwRoqccE5o7lOHDT12ttsrPIbybTvyNm+Pl7ZAwbWYtYRS7EPHICAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afb13a3fec53ea184fe252b09b9e80a25ca0ec2a4c5b1d851d248ab862df149b","last_reissued_at":"2026-05-17T23:53:13.096840Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:13.096840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Illumination of convex bodies with many symmetries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Konstantin Tikhomirov","submitted_at":"2016-06-29T06:47:18Z","abstract_excerpt":"Let $n\\geq C$ for a large universal constant $C>0$, and let $B$ be a convex body in $R^n$ such that for any $(x_1,x_2,\\dots,x_n)\\in B$, any choice of signs $\\varepsilon_1,\\varepsilon_2,\\dots,\\varepsilon_n\\in\\{-1,1\\}$ and for any permutation $\\sigma$ on $n$ elements we have $(\\varepsilon_1x_{\\sigma(1)},\\varepsilon_2x_{\\sigma(2)},\\dots,\\varepsilon_nx_{\\sigma(n)})\\in B$. We show that if $B$ is not a cube then $B$ can be illuminated by strictly less than $2^n$ sources of light. 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