{"paper":{"title":"Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Nam-Kiu Tsing, Pan-Shun Lau, Tuen-Wai Ng","submitted_at":"2016-08-22T09:51:37Z","abstract_excerpt":"Let $A\\in \\mathbb{R}^{N\\times N}$ and $\\mathrm{SO}_n:=\\{ U \\in \\mathbb{R}^{N \\times N}:UU^t=I_n,\\det U>0\\}$ be the set of $n\\times n$ special orthogonal matrices. Define the (real) special orthogonal orbit of $A$ by \\[ O(A):=\\{UAV:U,V\\in\\mathrm{SO}_n\\}. \\] In this paper, we show that the linear image of $O(A)$ is star-shaped with respect to the origin for arbitrary linear maps $L:\\mathbb{R}^{N\\times N}\\to\\mathbb{R}^\\ell$ if $n\\geq 2^{\\ell-1}$. In particular, for linear maps $L:\\mathbb{R}^{N\\times N}\\to\\mathbb{R}^2$ and when $A$ has distinct singular values, we study $B\\in O(A)$ such that $L(B)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06101","kind":"arxiv","version":1},"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"}