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We prove the following exact inequalities for any convex body $K\\subset\\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\\subset \\mathbb{R}^n$: \\begin{align*} &V_i \\big( K\\cap E \\big) \\geq \\left( \\frac{i+1}{n+1} \\right)^i \\max_{x\\in K} V_i \\big( ( K-x) \\cap E \\big) , \\\\ &\\widetilde{V}_i \\big( K\\cap E \\big) \\geq \\left( \\frac{i+1}{n+1} \\right)^i \\max_{x\\in K} \\widetilde{V}_i \\big( ( K-x) \\cap E \\big) ; \\end{align*} $V_i$ is the $i$th intrinsic volume, and $\\widetilde{V}_i$ is the $i$th dual volume taken within $E$. Our results are"},"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":"1809.05775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-15T21:57:15Z","cross_cats_sorted":[],"title_canon_sha256":"6357c489f463fc179804c59b50521e700340405daf21c9a5897ae65c936561ca","abstract_canon_sha256":"bb05edc81c6c1e519973f332c15bd5323eec98c1ff4bd2bbec8be1fd23b887c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:36.530070Z","signature_b64":"qbczqepMLL5dwG0CUsGhWTshPbImTARE4zc84OyXsLIlfQCgpnvN3VEfrm9EkK9v2QXuC9X9HJD3f8NoBcPCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6a8a4900899f73e9edd1f49c057b4a3db116e681ac0db1a11bdefdcac8375c4","last_reissued_at":"2026-05-18T00:05:36.529603Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:36.529603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Applications of Gr\\\"unbaum-type inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Matthew Stephen, Vladyslav Yaskin","submitted_at":"2018-09-15T21:57:15Z","abstract_excerpt":"Let $1\\leq i \\leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\\subset\\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\\subset \\mathbb{R}^n$: \\begin{align*} &V_i \\big( K\\cap E \\big) \\geq \\left( \\frac{i+1}{n+1} \\right)^i \\max_{x\\in K} V_i \\big( ( K-x) \\cap E \\big) , \\\\ &\\widetilde{V}_i \\big( K\\cap E \\big) \\geq \\left( \\frac{i+1}{n+1} \\right)^i \\max_{x\\in K} \\widetilde{V}_i \\big( ( K-x) \\cap E \\big) ; \\end{align*} $V_i$ is the $i$th intrinsic volume, and $\\widetilde{V}_i$ is the $i$th dual volume taken within $E$. 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