After reading the papers of Erdös and Szekeres regarding the "Happy Ending problem", I am wondering if there is some research / paper regarding the existence of a given number of points $F_2(n)$ in general position such that every subset of $n$ points is a convex n-gon.
For instance, $F_2(3)=3$, as every set of three points forms a triangle, which is always convex. I am not sure if $F_2(n>3)$ does even exist, I have not found an example of such a set of points. I guess that it does not exist, but I am not sure of how it would be possible to prove it.
If we generalize the question to three dimensions, regarding the existence of a given number of points $F_3(n)$ in general position such that every subset of $n$ points is a convex polyhedron, I get that $F_3(4)=4$, because every set of four points in general position in three dimensions form a convex polyhedron.
Thus, it can be conjectured that, for a given $d$ number of dimensions, the only given number of points $F_d(n)$ such that every subset of $n$ points is a given convex polytope is $F_d(n=d+1)=d+1$. Also, it seems trivial that necessarily $F_d(n<d+1)$ does not exist, as it is impossible to form any convex $d$-polytope with $d$ or less points.
If you could provide some research / paper regarding this question, on how to prove the conjecture, or provide an example of some $F_d(n\neq d+1)$, it would be welcomed.
Thanks in advance!
It seems to me that the answer is no: in the plane take three points not on a line and a fourth point inside the triangle. No matter how many points you add, the initial four points do not form a convex 4-gon.
This generalizes to all dimensions.