Let $\mathcal{K}$ be a compact convex set in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{K}}$. 9a. Show that the set of $\lambda \geqslant 0$ such that $-\lambda \mathcal{K} \subset \mathcal{K}$ is an interval. We denote $$a(\mathcal{K}) = \sup\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$$ 9b. Show that $a(\mathcal{K}) < \infty$ and that $a(\mathcal{K}) = \max\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$. 9c. Show that $0 < a(\mathcal{K}) \leqslant 1$. Deduce that $a(\mathcal{K}) = 1$ if and only if $\mathcal{K}$ is symmetric with respect to 0.
Let $\mathcal{K}$ be a compact convex set in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{K}}$.
9a. Show that the set of $\lambda \geqslant 0$ such that $-\lambda \mathcal{K} \subset \mathcal{K}$ is an interval.
We denote
$$a(\mathcal{K}) = \sup\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$$
9b. Show that $a(\mathcal{K}) < \infty$ and that $a(\mathcal{K}) = \max\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$.
9c. Show that $0 < a(\mathcal{K}) \leqslant 1$.
Deduce that $a(\mathcal{K}) = 1$ if and only if $\mathcal{K}$ is symmetric with respect to 0.