Do projections of convex sets equal (up to an affine transformation) some intersection with a hyperplane?

Question

Let $C$ be a convex subset of $\mathbb{R}^{n}$ and $C'$ its projection into a k-subspace $H\subseteq\mathbb{R}^{n}$ for $k\leq n.$ We can suppose for simplicity $p\colon\mathbb{R}^{n-k}\times\mathbb{R}^{k}\to\mathbb{R}^{k},(x,y)\mapsto y$ with $p(C)=C'.$ Does there exist in general a k-plane $H'\subseteq\mathbb{R}^{n}$ such that $C'$ is affinely (or rigidly, that is, after a rigid transformation) equivalent to $H'\cap C$?

Answer 1

Not always. Think of a thinly cut slanted piece of salami. Some projection of it will be a disk, but there may be no slice of it that is a disk.

${\bf Added:}$ Here is a better example. Consider the intersection of two cylinders

The projection of one of the blue pieces on a plane perpendicular to an axis of one of the cylinders is a disk. However, no plane section of such a piece is an ellipse.