Exercise 16, § 5

Let A be a compact set with interior points in $\mathbf{R}^n$. Show that if each frontier point of A lies on at least one support hyperplane of A, then A is convex. (Obtain a contradiction, showing that if x and y are two points of A such that the segment with end points x, y is not contained in A, and if z is an interior point of A not situated on this segment, then there exists a frontier point of A, distinct from x and y in the triangle with vertices x, y, z.)

EXERCISES