Let S beany non empty subset of anaffine space E. Given some point a∈S, we say that S is star-shaped with respect to a if the line segment [a,x] is contained in S for every    For all λ such that   We...

Let S beany non empty subset of anaffine space E. Given some point a∈S, we say that S is star-shaped with respect to a if the line segment [a,x] is contained in S for every

For all λ such that
  We say that S is star-shaped if it is star-shaped w.r.t. to some point a∈S.

(1) Prove that every non empty convex set is star-shaped.

(2) Show that the re are star-shaped subsets that are not convex. Show that the rear en on empty subsets that are not star-shaped (give an example in An, n = 1,2,3).

(3) Given a star-shaped subset S of E, let N (S) be these to fall points a∈S such that S is star-shaped with respect to a. Prove that N(S) is convex.