 Solution

For $\Sigma =I_d$, $\mathrm{tr}(\Sigma )=d$ and $\Vert \Sigma {\Vert }_2=1$. Taking $k$ of order $\max \{nϵ,1\}$ in part d givesFor Gaussian data under Huber contamination, the Tukey median has the sharper dependenceup to universal constants in its valid contamination range. Both have the same sampling term $\sqrt{d/n}$, but the Tukey median is linear rather than square-root in the contamination fraction.