{"paper":{"title":"Answer to a question of Kolmogorov","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Andr\\'as M\\'ath\\'e, M\\'arton Elekes, Rich\\'ard Balka","submitted_at":"2012-10-21T19:54:55Z","abstract_excerpt":"More than 80 years ago Kolmogorov asked the following question. Let $E\\subseteq \\mathbb{R}^{2}$ be a measurable set with $\\lambda^{2}(E)<\\infty$, where $\\lambda^2$ denotes the two-dimensional Lebesgue measure. Does there exist for every $\\varepsilon>0$ a contraction $f\\colon E\\to \\mathbb{R}^{2}$ such that $\\lambda^{2}(f(E))\\geq \\lambda^{2}(E)-\\varepsilon$ and $f(E)$ is a polygon? We answer this question in the negative by constructing a bounded, simply connected open counterexample. Our construction can easily be modified to yield the analogous result in higher dimensions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5758","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}