$n$ points forming a convex $n$-gon Suppose I am given a collection of $n$ points, any four of which form a convex quadrilateral. I wish to establish that these $n$ points form a convex $n$-gon.
I am thinking about using induction. The case $n=4$ is trivial. If the result is assumed for $n-1$ how do I establish it for $n$?
Thanks.

Consider the convex hull. If it has fewer than $n$ points, then there's a point in its interior. Use that point to form a non-convex quadrilateral - do you see a way to do this?
