Exercise 1.18 Consider a closed simple curve C in the plane (a curve which does not auto-intersect) with the following property:

Exercise 1.1.18 Consider a closed simple curve C in the plane (a curve which does not auto-intersect) with the following property: For any point A C, the tangent line through A to C leaves the curve C on the same half-plane. Then the set K = Int C is convex. Is the converse true? Definition 1.1.19 The interior of an angle BAC is the intersection of the halfplanes HB and HC , where HB is the half-plane determined by AC and B and HC is the half-plane determined by AB and C. As the intersection of two convex sets is a convex set, it follows that the interior of an angle is a convex set. Exercise 1.1.20 Define the interior of a triangle. Prove that it is a convex set.