While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here.

Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, $P\cap X$ is simply connected. Is $X$ convex?

I'm not sure, but I think maybe it's necesary to assume some well behaving like local simple connectedness. Anyway I think this is true with the apropiate asumptions. I would not be surprised if it was true just as stated.

Probably this is true even in greater dimensions.