Suppose we have a closed, bounded, convex set $K \subset \mathbb{R}^n$ with non-empty interior. It's well-known that we can approximate $K$ either from the inside or from the outside in the Hausdorff distance either by closed convex polytopes or by closed convex sets with smooth boundaries. That, is, we can find approximates $K_m$ such that either $K_m \subset K$ for all $m$, or $K \subset K_m$ and $K_m \to K$ in the Hausdorff metric.
I've encountered the following variant of this problem. The set $K$ now has a "flat part" on its boundary. That is, there is a hyperplane $\Gamma \subset \mathbb{R}^n$ and a relatively open set $\varnothing \neq S \subset \partial K$ such that $S \subset \Gamma$. Some examples would be the usual hypercube and the half-ball. Is it possible to approximate $K$ from one side (either one will do, but I'm more concerned with outside) by a sequence $\{K_m\}_{m=1}^\infty$ of closed convex sets with $C^2$ (twice continuously differentiable) boundary in such a way that $S \subset \partial K_m$ for all $m$? In other words, can we do the approximation in such a way that all of the approximates share the prescribed "flat part"?
I can make this work in $\mathbb{R}^2$ by first approximating from the outside by polygons, which can be forced to share the flat part, and then approximating the polygons from the inside by convex sets with smooth boundaries. To pull off the last step I can use a standard mollification trick to "smooth out" the vertices of the polygon in a way that makes it easy to glue together the smoothed curves.
This gluing procedure becomes a nightmare for $\mathbb{R}^n$ with $n \ge 3$, and I don't know how to proceed. After digging around in some books on convexity, it's not clear to me that any of the approximations procedures preserve the flat part. If anyone has an idea for how to attack this or can point me toward a reference that has an approximation scheme that is compatible with preserving flat parts, I would be grateful to see it.
Thanks!
I eventually found what I was looking for in this paper: M. Ghomi. Optimal smoothing for convex polytopes. Bull. London. Math. Soc. 36 (2004), 483-492.