HW4

Problem 1

• 1. 
     A star is an example. P(g1) can see all of P, but on all direction of the circle, it is not monotone.
• 2. Same as 3. 3 is a subset of 2.
• 3. 
     This is an example. It is a orthogonal simple polygon, and it is montain monotone on X direction, but it is not one gaurdable(two witness points do not share any common view).
• 4. 
• 5. True. Simple 4-gon with exactly 3 convex vertices must be a fox. If we rotate it and put its nose on origin and one of its ear on positive x-axis, it will look something like this:
     Note that if we go through y direction, the nose must be t because its at (0,0) which is the lowest y coordinate. Also the ear that is not on the x-axis must be b because it has highest y coordinate or otherwise this polygon won't be a fox. Besides, other than the line at $y=0$, no more than two points will share the same y value. Thus we have t and b join the same line so this must be mountain monotone on some direction d.

Problem 2