Is the set of convex bodies include in a closed ball compact?

Question

I consider the set $\mathcal{K}_B$ of convex bodies (convex and compact) which are include inside the unit closed ball of $\mathbb{R}^d$. I endow this set with the Hausdorff distance. Is it compact?

Answer 1

Let ${\mathcal C}_B$ denote the space of closed subsets of $B$ equipped with Hausdorff metric. Then ${\mathcal C}_B$ is compact, see e.g. here. Then you check by a direct argument that ${\mathcal K}_B$ is a closed subset of ${\mathcal C}_B$ (limit of a sequence of convex subsets is convex). Now, argue that a closed subset of a compact metric space is compact.

Edit: As requested, here is a proof of convexity of the limit. Suppose that $C_n$ is a sequence of closed convex subsets Hausdorff-converging to a closed subset $C\subset B$. To verify that $C$ is convex, take two points $p, q\in C$; then $p=\lim_{n\to\infty} p_n, q=\lim_{n\to\infty} q_n$ for sequences $p_n\in C_n, q_n\in C_n$. Then the sequence of affine maps $$ f_n: [0,1]\to C_n, t\mapsto (1-t)p_n+ tq_n $$ converges uniformly to an affine map $$ f: [0,1]\to B, t\mapsto (1-t)p+ tq. $$ In particular, the sequences of the images of $f_n$'s Hausdorff-converges to the image of $f$ (this is a general fact about Hausdorff-convergence). Thus, the image of $f$ is contained in $C$. It follows, that $C$ contains the interval spanned by $p, q$, hence, $C$ is convex. qed