Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits

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 $L:\mathbb{R}^{N\times N}\to\mathbb{R}^2$.