how to show that convex hull of a given set of points has the smallest area (Solved)

Question

How to prove this: The minimum area convex polygon enclosing a set of points is the convex hull of the points.

Answer 1

From the comments, your definition is: The convex hull of a set of points S is the set of all convex combinations of points of S. Define the minimum area convex polygon enclosing S as $A$ and the convex hull of S as $C$.

1. $A$ is convex and contains the points in $S$, so $A$ must contain the convex combinations of the points in $S$. Can you conclude that $C \subseteq A$?
2. Both $A$ and $C$ are convex and enclose the points of $S$, so $A \cap C$ is convex and encloses the points of $S$. What does that mean about $A \cap C$ if $A$ has minimum area? In particular, can you conclude that $A \subseteq C$?

Answer 2

Hint: The convex hull may be defined as the intersection of all convex sets containing the given set $S$. So the convex hull is contained in any convex polygon containing $S$.