Question

Conv{...} means the convex hull of these points. In question (b), the convex hull is the area of the square formed by the four points.

Let A CRd. The boundary of A is the set A= {1ER : Br(2) NA #0 and Br(2) NA #0 for all r >0}. In other words, a point r e Rd is in the boundary of A if and only if every ball centered at z intersects both A and A. (a) What is the boundary of an interval (a,b) in Ri? Does it matter if the endpoints are included in the interval? (b) Let A = conv {(1,0),(-1,0),(0,1),(0, -1)}. What is a A? Let A CRd. Show that A is closed if and only if A CA.

Answer 1

=(a) Boundarry of (a,b) is {a,b}. It end points are included to the interval then, it does not metter. Because Bolal n [a, b] and Bg(a) n{[a,b]}" is arre non empty. Also for the same for point be The bold region is boundarry. OA={x: x=1-4, 1565o} U{ x: n=-1-y, ley=0} V{x: x= 1+y, tsy so} U{ *:*= y-t, oxysi} Tot oo A is closed lify DACA Proof first assume A is closed. let the re exirt a point de ONAJ such that n&A . Then REA since A is closed then AC is open so there exif a boll Bg(x) such that Bolu C AC ie Bolu) does not contains any point of A - which is a contradiction because X EQA - so DACA. conversely, assume that aCASA . To poove A is open, we assume that A is oken &XEA, there exist oxo such that Br(W) C Á. But then that I ball contains no points in AC, so can not be boundarry point of A. which is contradiction that O DA CA. I