Let f mapping R into R be a function, and let X be a subset of R2. (a) Define what it means for f to be convex. (b) Define what it means for X to be convex. (c) Define epi(f) = {(x,y)|y>=f(x)}. Describe the relationship between epi(f) and the graph of y = f(x) in R2. (d) Prove that if epi(f) is a convex set, then f is a convex function.

Let f mapping R into R be a function, and let X be a subset of R².

(a) Define what it means for f to be convex.

(b) Define what it means for X to be convex.

(c) Define epi(f) = {(x,y)|y>=f(x)}. Describe the relationship between epi(f) and the graph of y = f(x) in R².

(d) Prove that if epi(f) is a convex set, then f is a convex function.