Note 2, DM819, fall 2018

Exercises September 10

• Exercises (1.1, 1.2,) 1.3, 1.4.a, 1.7, 1.8, 1.9.

  Exercises 1.1 and 1.2 are about proving that the intersection of all convex sets containing P equals the smallest-perimeter polygon with vertices from P containing P. A better sequence of small proofs is the one below, which we will carry out instead of following the outline of Exercises 1.1 and 1.2. Prove that

  1. the intersection of a collection of convex sets is again convex.
  2. any smallest-perimeter polygon containing P has vertices only from P.
  3. any smallest-perimeter polygon containing P is convex.
  4. any smallest-perimeter polygon containing P is contained in any convex set containing P.
  5. any smallest-perimeter polygon with vertices from P equals the intersection of all convex sets containing P (making the polygon unique).
  Many of the arguments are most easily carried out as proofs by contradiction.

  In 1.8.a, you may assume that the two convex hulls are separated by a vertical line.

Lecture September 11

• Polygon Triangulation (chapter 3).