Exercises
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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
- the intersection of a collection of convex sets is again convex.
- any smallest-perimeter polygon containing P has vertices only from P.
- any smallest-perimeter polygon containing P is contained in any convex set containing P.
- any smallest-perimeter polygon containing P is convex.
- 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.