Let X=R^2 and the distance be the usual Euclidean distance. If C and D are non-empty sets of R^2 and we have:

C+D := {y ϵ R^2 | there exists c ϵ C and dϵD s.t c+d = y}

A) What is C+D if the open balls are C= ball((0.5,0.5);2) and D=ball((0.5,2.5);1)

B) Same as A) expect D is now a closed ball

C) same as a) except D={(l,-1)|l ϵ R}

D) Is the following true? If C,D are non-empty subsets of R^2 s.t C is open, then the sum C+D is open.

I'm assuming "ball((a,b);r)" refers to the open balls of radius r around the point (a,b).

(A) C+D = ball( (1,3) ; 3 )

(B) C+D = ball( (1,3) ; 3 )

(C) C+D = { (x,y) ϵ R^2 | -2.5 < y < 1.5 }

(D) Yes. Essentially, C+D is the union of a bunch of translated copies of C, each of which is open.