I was looking over a chapter in my textbook on linear programming, when I thought of something interesting, but probably unrelated.
On a plane, many straight lines - each one extended to infinite length in both directions - are drawn. Let $x$ be a point (the red dot in the diagram below) in the plane so that it is inside (or on the boundary of) a polygon formed by parts of the straight lines. Then the smallest polygon - in terms of area - that contains $x$ (i.e. the green polygon) is convex!
Why is this?
I don't think we have a convex hull to work with in an attempted proof.
Hmmm... if you join up all the intersection points of the lines then you just get lots of triangles. It’s not obvious how this helps though. $$$$
Note that this doesn't work if the lines themselves are strictly convex (or concave), e.g.:
The green shape is not convex because the straight line joining the red dots does not lie inside the green shape (this is the definition of convex).
Your lines are actually half-spaces, splitting the plane into the half that contains the point (and excluding the other half), at each line.
Because the intersection of a finite number of half-spaces is convex, the polygon around the dot is convex.