Proving Convexity of Set S with Concave gi Functions

[SNOOTCHIEBOOCHEE]

Homework Statement

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

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 I am stuck.

 

[SNOOTCHIEBOOCHEE]

any thoughts?

 

[Billy Bob]

How do you show a set is convex?

Can you do this when m=1?

 

[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 don't know how to use that here.

 

[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.