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He further asks whether it is possible to subdivide $M$ by a disk $D$ into two regions of volume $V/4$ so that the area of $D$ is bounded by some function $h(d,A,V)$.\n  In this paper, we answer the questions above in the negative. We further prove that given $N>0$ and $c\\in(0,1)$, one can construct a metric $g'$ so that any 2-disk $D$ su"},"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":"1508.03746","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-08-15T15:59:47Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"73c334db8e20cc18345e24456a0f3b43e78fe12bbe66dcc20be9ce416eff07fc","abstract_canon_sha256":"d96b8291fe191b53639f669312a916768a036429a0d425474a8abc8897d58789"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:09.894786Z","signature_b64":"fq7zHAmaxez3OeapsgloeQFco7eK+76Wh6JljfVNkMIPTPGQsqGeEU/rhgG99FMImK0EgnQUl+9ThBduD+LvDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aca7417d0449c301b1c26c4a569ce3257a9fe22c454cda1a48fd5912b5918dcf","last_reissued_at":"2026-05-18T00:50:09.893974Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:09.893974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subdividing Three-Dimensional Riemannian Disks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Parker Glynn-Adey, Zhifei Zhu","submitted_at":"2015-08-15T15:59:47Z","abstract_excerpt":"P. Papasoglu asked in [Pap13] whether for any Riemannian 3-disk $M$ with diameter $d$, boundary area $A$ and volume $V$, there exists a homotopy $S_t$ contracting the boundary to a point so that the area of $S_t$ is bounded by $f(d,A,V)$ for some function $f$. He further asks whether it is possible to subdivide $M$ by a disk $D$ into two regions of volume $V/4$ so that the area of $D$ is bounded by some function $h(d,A,V)$.\n  In this paper, we answer the questions above in the negative. 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