# Convex region

1. Mar 11, 2017

### matrixone

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

2. Relevant equations

3. The attempt at a solution
Consider S = {(1,1)} and T = {(0,0)}
Clearly, S and T is convex
S + T = S and S - T = S
So both of them are convex.

But i feel that the answer is too simple....and seems that i wrongly interpreted the question ....
Any thoughts?

2. Mar 11, 2017

### Staff: Mentor

What if you choose $T=\{(2,2)\}\,$? I assume the statement has to be true for any choice of $S,T$. And it didn't say, that they have to be convex. More interesting is if $S,T$ are balls (or squares, which are easier to handle in this case) of different radius and centers, what can be said then? Can this be generalized to convex sets with a non-empty interior?

3. Mar 11, 2017

### matrixone

In the last line of the question, "Which of these is TRUE FOR ALL CONVEX SETS S & T? "

And, even if for only one choice of vertices, S & T both are convex, other options can be eliminated. Right ?

4. Mar 11, 2017

### Staff: Mentor

Sorry, overlooked. NO NEED TO USE CAPS.
An example isn't a proof. If you eliminate choices, you have to prove that it can be done without restricting the general case.

5. Mar 11, 2017

### Dick

Actually the question does say "Which of the following is TRUE for all convex sets S and T?". I would suggest that you jump straight to trying to prove E... Might save you some time over looking at lots of a examples.