I am trying to figure out how to solve:

$\min_U r_{p}$

where $r_{p}=\alpha^\intercal\omega$ and $U$ is a sphere centered at $\alpha$ with radius equal to $\chi|\alpha|$ . ( $\omega$ is a vector or weights and $\chi$ lies between 0 and 1.)

The authors end up with the following solution:

$\min_U r_{p}=\alpha^\intercal\omega-\chi|\alpha||\omega|$

The authors hint at Bayesian estimation but I am not familiar with it. Any ideas as to how they may have arrived at this would be great. Many thanks in advance.