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# 代写经济学问题集 Economics 426代写 经济学作业代写

161

## Economics 426: Problem Set 6– Convexity and Separation

Due: Wednesday, April 8, 2:00pm online.

Points Possible: 100.

### I. 代写经济学问题集

Suppose F(x, y) := x1/2 y1/2 and g(x, y) = x2 + y2 .

A. Draw A := {(x, y)|g(x, y) 1} and B := {(x, y)|F(x, y) > 1}. (8 pts)

B. Are A and B convex? A well-drawn figure is sufficient. (8 pts)

C. Are A and B strictly convex? A well-drawn figure is sufficient. (8 pts)

D. Are A and B disjoint? Explain. (8 pts)

E. Can A and B be separated by a hyperplane? Explain. If yes, provide such a hyperplane. (8 pts)

### II.

Show that the budget set B := {(x, y)|x + 2y ≤ 4, x, y ≥ 0} is convex, but not strictly convex. (9 pts)

### III. 代写经济学问题集

Show that if a set A ⊂ Rn is open and convex, then it must be strictly convex. (8 pts)

A. Show by (counter)example (a well-drawn figure is sufficient) that the convexity of both A and B are typically required to ensure this result. That is, show that if either of A or B is nonconvex then there may be no separating hyperplane. (8 pts)

B. Construct an example of disjoint non-convex that can be separated. Does this contradict the separating hyperplane theorem? Explain. (8 pts)

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