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

TL;DR

This paper investigates the geometric properties of linear images of special orthogonal orbits, proving star-shapedness under certain conditions and providing insights into convexity for specific cases.

Contribution

It establishes star-shapedness of linear images of special orthogonal orbits for high-dimensional linear maps and offers an alternative proof of convexity results in two dimensions.

Findings

Linear images are star-shaped if the orbit dimension condition is met.

For 2D linear maps, the boundary points of the image are characterized.

Provides an alternative proof of convexity of the orbit image in 2D.

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