Homework #1
Assigned: Jan. 3rd, 2012
Due: Jan. 10th, 2012
Points: 40
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

The two questions in this homework require you to solve two problems that are dual to each other using the statements given below. So, you will be solving four problems altogether. You have to pose the four prolems and solve them using what you know about minimizing/maximizing using calculus. You may use Matlab if you wish. Write the problem statements in the manner we discussed in the first lecture. Write down the solutions properly. You may use figures if you wish.
This homework is to know what all you know about finite variable optimization!

1. 20 points
   The shortest distance from a point to a convex shape in 2D not containing the point is equal to the maximum of the distances from the point to a line separating the shape and the point.
   Take the shape to be a circle of radius 1 centred at the origin. Let the point be located at (2,3) if you want to solve it numerically. Symbolic solution is preferred.
2. 20 points
   The altitude of the largest equilateral triangle circumscribing a given triangle is equal to the minimum sum of the three distances from a point inside the triangle to the vertices of the given triangle."
   If you want to solve the problem numerically, let the vertices of the given triangle be (0,0), (4,4), and (-3,5).