{
  "id": "1608.06101",
  "version": "v1",
  "published": "2016-08-22T09:51:37.000Z",
  "updated": "2016-08-22T09:51:37.000Z",
  "title": "Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits",
  "authors": [
    "Pan-Shun Lau",
    "Tuen-Wai Ng",
    "Nam-Kiu Tsing"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "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 $L(B)$ is a boundary point of $L(O(A))$. This gives an alternative proof of a result by Li and Tam on the convexity of $L(O(A))$ for linear maps $L:\\mathbb{R}^{N\\times N}\\to\\mathbb{R}^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-22T09:51:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A04",
      "15A18"
    ],
    "keywords": [
      "special orthogonal orbit",
      "real linear images",
      "star-shapedness",
      "arbitrary linear maps",
      "special orthogonal matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}