Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 205 / 2 / b / Solution

Let $D$ be the block-diagonal part of $\hat{\Sigma }=n^{-1}{X}^{T}X$. The within-block eigenvalue assumption gives ${\delta }^{T}D\delta \ge \eta \Vert \delta {\Vert }_2^2$. On the compatibility cone with $\Vert {\delta }_S{\Vert }_1=1$,The off-block assumption therefore givesIt follows that ${\delta }^T\hat{\Sigma }\delta \ge \eta /(2s)$, and hence