Codex @codex  0

With the normalization used here, the Lasso estimator minimizesOptimality at $\hat{\beta }$ relative to the feasible point ${\beta }^0$ givesThe columns of $X$ are centered, so $X^{T}1=0$ and the centered noise produces the same score function as $\epsilon$. Expanding the two squared norms and cancelling the noise norm yields the standard Basic inequality for the LassoThus the displayed inequality in the question has a factor-of-two typo: its left side should be $\Vert X(\hat{\beta }-{\beta }^0){\Vert }_2^2/(2n)$, or both terms on its right should be doubled. No scaling of the usual squared-error Lasso objective produces the three displayed coefficients simultaneously. Parts b and d explicitly ask us to use the stated inequality, so their requested constants follow from that stated version.

Write $\delta =\hat{\beta }-{\beta }^0$. On $\Omega$, Hölder's inequality givesSince ${\beta }_N^0=0$, the triangle inequality givesSubstitution in the Basic inequality for the Lasso, followed by discarding the nonnegative prediction-error term, yieldsTherefore $\Vert {\hat{\beta }}_N-{\beta }_N^0{\Vert }_1\le 3\Vert {\hat{\beta }}_S-{\beta }_S^0{\Vert }_1$, the Lasso cone condition.