We denote $\|\cdot\|_2$ the canonical Euclidean norm on $\mathbb{R}^2$ and we denote $$\mathcal{C} := \left\{ x \in \mathbb{R}^2 \mid \|x\|_2 = 1 \right\}$$ We fix an arbitrary norm $\|\cdot\|$ on $\mathbb{R}^2$ and we denote $$\mathcal{K} = \left\{ A \in M_2(\mathbb{R}) \mid \forall x \in \mathbb{R}^2,\ \|x\|_2 \geq \|Ax\| \right\}.$$ a) Show that $\mathcal{K}$ is a compact and convex subset of $M_2(\mathbb{R})$. b) Show that there exists $A \in \mathcal{K}$ such that $\det A = \sup_{B \in \mathcal{K}} \det B$.
We denote $\|\cdot\|_2$ the canonical Euclidean norm on $\mathbb{R}^2$ and we denote
$$\mathcal{C} := \left\{ x \in \mathbb{R}^2 \mid \|x\|_2 = 1 \right\}$$
We fix an arbitrary norm $\|\cdot\|$ on $\mathbb{R}^2$ and we denote
$$\mathcal{K} = \left\{ A \in M_2(\mathbb{R}) \mid \forall x \in \mathbb{R}^2,\ \|x\|_2 \geq \|Ax\| \right\}.$$
a) Show that $\mathcal{K}$ is a compact and convex subset of $M_2(\mathbb{R})$.\\
b) Show that there exists $A \in \mathcal{K}$ such that $\det A = \sup_{B \in \mathcal{K}} \det B$.