Convexity and star-shapedness of real linear images of special orthogonal orbits

Let A∈Rn×nA∈Rn×n and SOn:={U∈Rn×n:UUt=In,detU>0}SOn:={U∈Rn×n:UUt=In,detU>0} be the set of n×nn×n special orthogonal matrices. Define the (real) special orthogonal orbit of A byO(A):={UAV:U,V∈SOn}.O(A):={UAV:U,V∈SOn}. In this paper, we show that the linear image of O(A)O(A) is star-shaped with respect to the origin for arbitrary linear maps L:Rn×n→RℓL:Rn×n→Rℓ if n≥2ℓ−1n≥2ℓ−1. In particular, for linear maps L:Rn×n→R2L:Rn×n→R2 and when A   has distinct singular values, we study B∈O(A)B∈O(A) such that L(B)L(B) is a boundary point of L(O(A))L(O(A)). This gives an alternative proof of a result by Li and Tam on the convexity of L(O(A))L(O(A)) for linear maps L:Rn×n→R2L:Rn×n→R2.