TITLE: A characterization of convexity
QUESTION [15 upvotes]: 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.

REPLY [17 votes]: A version of this is well known:

Let $X\subset \Bbb R^3$ be a compact
  set and suppose every intersection of
  $X$ by a plane is contractible. Then
  $X$ is convex.

This is due to Schreier (1933) in $\Bbb R^3$, and Aumann (1936) generalized this to higher dimensions.  See this and other related results in Ju.D. Burago and V.A. Zalgaller, Sufficient criteria of convexity, J. Math. Sci. 10 (1978). 395–435.  Incidentally, this a (hard) exercise in my book (Exc. 1.25).