Let $C$ be a compact convex subset of Euclidean space. Recall that $x\in C$ is an *exposed point* of $C$ if there is a plane $P$ such that $P\cap C = \{x\}$. It is obvious that exposed points are extreme. To see that they are not equivalent, draw two parallel lines and put half circles on each end.

Straszewicz's theorem asserts that the exposed points are dense in the extreme points. In particular, the closed convex hull of the exposed points of $C$ is equal to $C$.

I would like a quantitative version. Most likely there is a well-known quantitative version, but I am googling the wrong words.

Here is a proposed version:

Call a point $x\in C$ an *$\epsilon$-exposed point of $C$* if there is a ball $B$ of radius $r\leq \epsilon^{-1}$ such that $C \subseteq B$ and $C \cap \partial B = \{ x \}$. Let $C_\epsilon$ be the closed convex hull of the $\epsilon$-exposed points of $C$. Then the Hausdorff distance between the sets $C$ and $C_\epsilon$ is at most $A \epsilon$ (or something else, but I'd like it explicit in $\epsilon$) where $A>0$ depends on the dimension and the diameter of $C$.

(I am not sure this is very difficult and I am not sure this is very easy. Just wondering if something like this already exists before I am forced to spend time proving it...)