Would number of vertex of convex hull of subset smaller than the set?

Question

For a finite discrete set with $n$ elements $X \subset \mathbb{R}^2$, one can find its Convex Hull $CH(X)$ and the vertices of Convex Hull.

Define a function $f$ mapping from the set to the number of vertices on the convex hull: $f(X)$.

Is this claim always true ?

$f(X) \geq f(S), \forall S \subset X$

Answer 1

the claim is false.

E.g. the set can be a circle enclosed in the square.

the vertices on the square are going to be the vertices on the convex hull. the circle is clearly a subset, which clearly have more vertices on the convex hull.