Exercise 1, § 2

A subset A of a vector space E, is starshaped relative to 0 if for all $x \in A$ and every $\lambda$ such that $0 \leq \lambda < 1$, the point $\lambda x$ belongs to A. Let A be starshaped and such that, for each $x \in A$, there exists $\mu > 1$ such that $\mu x \in A$. Show that if, for every pair of points $x, y$ of A we have $\frac{1}{2}(x + y) \in A$, then A is convex. Give an example of a non-convex starshaped set A such that $\frac{1}{2}(A + A) \subset A$.