How do I find $\min_{z,z'~s.t. \|z\|=\|z'\|=1} \|A^Tz-B^Tz'\|$ where $A$ and $B$ have length 1 columns?

Question

For

• $n\times m$ matrices $A$ and $B$ ($n>>m$), which also have length 1 columns
• $rank(A)=rank(B)=m$
• $n$-dimensional vector $z$
• The $l2$ norm for vectors and Frobenius norm for matrices

I need to find $$\min_{z,z'~s.t. \|z\|=\|z'\|=1} \|A^Tz-B^Tz'\|$$

Do you think there is a closed solution? Any help would be greatly appreciated!