Homework Help: Proving Convexity

1. Apr 6, 2009

SNOOTCHIEBOOCHEE

1. The problem statement, all variables and given/known data

Let g1, ..., gm be concave functions on R^n . Prove that the set S={x| gi(x)$$\geq 0$$, i=1,...,m} is convex

3. The attempt at a solution

So i tried this using two different definitions.

First i used the definition that says f(y)$$\leq$$ f(x) + $$\nabla$$f(x)T(y-x)

then i substitued f(ax + (1-a)y)$$\geq$$ af(x) + (1-a)f(y)

and tried to do some manipulations to show that the inequalites wen the other way but that didnt come out right.

Now im stuck.

2. Apr 7, 2009

SNOOTCHIEBOOCHEE

any thoughts?

3. Apr 7, 2009

Billy Bob

How do you show a set is convex?

Can you do this when m=1?

4. Apr 7, 2009

SNOOTCHIEBOOCHEE

we can show a set is convex for for any elements x and y

ax + (1-a)y are in S. for a between 0 and 1. but i dont know how to use that here.

5. Apr 7, 2009

Billy Bob

OK, so what does it mean for x and y to be in S (this is the given)?

Again, do the m=1 case first, for simplicity.

When you get to it, just use f(ax + (1-a)y) $$\geq$$ af(x) + (1-a)f(y) for concave, not the other one.