Let $C$ be a compact set in $\mathbb{R}^n$. If $C$ is convex, it must be homeomorphic to a closed ball. Now suppose that instead of convexity we require the intersection of $C$ with any line to have at most $m$ connected components. What topological restrictions does this condition place on $C$ and its embedding in $\mathbb{R}^n$?
For example, the result at http://www.ics.uci.edu/~eppstein/junkyard/knot-colinear.html shows that for $n=3$ and $m=3$, the complement of $C$ can't be knotted.
I would be interested in results for specific small cases as well as in the following question: For fixed $n$, does there exist $m$ so that $C$ can come from any "reasonable" isotopy class? (Define reasonable reasonably.)