$\newcommand{\ones}{\mathbf 1}$

Let $x^\star$ be optimal for the least-norm problem \[ \begin{array}{ll} \mbox{minimize} & \|x\|_p \\ \mbox{subject to} & Ax=b, \end{array} \] with variable $x \in \mathrm{R}^n$, where $A\in \mathrm{R}^{m \times n}$, with $m \ll n$.

For $p=2$, we would expect to see many components of $x^\star$ equal to zero.

For $p=1$, we would expect to see many components of $x^\star$ equal to zero.

For $p=\infty$, we would expect many components of $x^\star$ to take on the values $\pm \|x^\star \|_\infty$.