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

Let $A\in {\mathbb{R}}^{N\times N}$ and $$\mathrm{S}\mathrm{O}$_n:={ U \in $\mathbb{R}$^{N \times N}:UU^t=I_n,\det U>0}$bethesetof$n\times n$ special orthogonal matrices. Define the (real) special orthogonal orbit of $A$ by $$O(A):=\{UAV:U,V\in {\mathrm{S}\mathrm{O}}_n\}\mathrm{.}$$ In this paper, we show that the linear image of $O(A)$ is star-shaped with respect to the origin for arbitrary linear maps $L:{\mathbb{R}}^{N\times N}\to {\mathbb{R}}^{\mathrm{\ell }}$ if $n\ge 2^{\mathrm{\ell }-1}$. 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}$^2$.