I am wondering about the following :

In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{C_k}(x) \rightarrow \mathbb{1}_C(x).$$ Is it true that the convex bodies $C_k$ converges to $C$ in the Hausdorff metric ? i.e. do we have, for any $\epsilon > 0$, a rank $k_0 \gg 1$ such that for any $k \geq k_0$, $C_k$ is included in the $\epsilon$-neighborhood of $C$ and conversely ?

Any comments/suggestions is welcome ! Thanks !

I am wondering about the following :

In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{C_k}(x) \rightarrow \mathbb{1}_C(x).$$

EDIT suppose also that you have $$\lim \int |\mathbb{1}_C-\mathbb{1}_{C_k}| = 0.$$

Is it true that the convex bodies $C_k$ converges to $C$ in the Hausdorff metric ? i.e. do we have, for any $\epsilon > 0$, a rank $k_0 \gg 1$ such that for any $k \geq k_0$, $C_k$ is included in the $\epsilon$-neighborhood of $C$ and conversely ?

Any comments/suggestions is welcome ! Thanks !