{"paper":{"title":"On Hausdorff dimension of radial projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2019-03-28T16:15:28Z","abstract_excerpt":"For any $x\\in\\mathbb{R}^d$, $d\\geq 2$, denote $\\pi^x: \\mathbb{R}^d\\backslash\\{x\\}\\rightarrow S^{d-1}$ as the radial projection $$\\pi^x(y)=\\frac{y-x}{|y-x|}. $$\n  Given a Borel set $E\\subset{\\Bbb R}^d$, $\\dim_{\\mathcal{H}} E\\leq d-1$, in this paper we investigate for how many $x\\in \\mathbb{R}^d$ the radial projection $\\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\\dim_{\\mathcal{H}}\\pi^x(E)=\\dim_{\\mathcal{H}} E$. We develop a general framework to link $\\pi^x(E)$, $x\\in F$ and $\\pi^y(F)$, $y\\in E$, for any Borel set $F\\subset\\mathbb{R}^d$. In particular, whether $\\dim_{\\mathcal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.12093","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"}