Let $M \in \mathcal{M}_{k \times d}(\mathbb{R})$ with $\operatorname{rank}(M) = k$, $b \in \mathbb{R}^k \setminus \{0\}$, and
$$r := \inf\left\{\|x\|_1, x \in \mathbb{R}^d, Mx = b\right\}.$$
Let $C$ be the set:
$$C := \left\{x \in \mathbb{R}^d : Mx = b, \|x\|_1 = r\right\}$$
Show that $C$ is non-empty, convex, closed and bounded.