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We show that a subspace $F\\in G_{n,n-k}$ of codimension $k=\\gamma n$, where $\\gamma\\in (1/\\sqrt{n},1)$, satisfies $$K\\cap F\\subseteq \\frac{c}{\\gamma }\\sqrt{n}L_K (B_2^n\\cap F)$$ with probability greater than $1-\\exp (-\\sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(\\mu )$ of an isotropic log-concave probability measure $\\mu $ on ${\\mathbb R}^n$. For every $1\\leq q\\leq n$ and $\\gamma\\in (0,1)$ we show that a random subspace $F\\in G_{n,(1-\\gamma )n}$ satisfies $Z_q(\\mu )\\cap F\\sub"},"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":"1601.02254","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-01-10T19:07:55Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"a2e5e8e6fbfb4182ed1c16f4b58a09df1fe30164b74437028599577dcd70b2df","abstract_canon_sha256":"4cb069e1427afab641215a971b383273453b7a65f280f1d4f9d815029340cc57"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:44.179335Z","signature_b64":"I5l/gA3/1D4+tXlyBIa3emebJlKj/0yEVHi/lsrOIgLuHo2RoM2RtAfk6F9tmc6Jj6nftTnFbN9ZaHxgGRwWAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"875f791cf63b43a08d696e48493bdc5e51a3809c2fc10dc72ecc13fb22c63cff","last_reissued_at":"2026-05-18T01:03:44.178712Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:44.178712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometry of random sections of isotropic convex bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Antonis Tsolomitis, Apostolos Giannopoulos, Labrini Hioni","submitted_at":"2016-01-10T19:07:55Z","abstract_excerpt":"Let $K$ be an isotropic symmetric convex body in ${\\mathbb R}^n$. 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