Random thoughts on the Lasso (to be continued...)

For any $n$-by-$p$ matrix $X$ (with non-zero rows), consider the following objects
$$\mathcal P_X := \{z \in \mathbb R^n | \|X^Tz\|_\infty \le 1\},\; D_X := \text{diag}(\|X_1\|_\infty,\ldots, \|X_n\|_\infty),\; Z_{D_X} := D_X^{-1}\mathbb B_1,$$
where $\mathbb B_1$ is the unit-ball for the $\ell_1$-norm.  Note that $Z_{D_X} \subseteq \mathcal P_X$.

Given a closed convex set $K \subseteq \mathbb R^n$, we've defined the euclidean projection
$$\text{proj}_K(a) := \text{the unique point of }K\text{ minimizing distance from }a \text{ to } K.$$
It's not (too) hard prove that $0 \le QP \le QA$.

What more (of geometric taste) can be said about the picture ?