Unsure of how to utilize definition of a convex set

Question

I have a set: [0,4] x [-2,3] and I am trying to determine its convexity.

I would like to use the definition of convexity: z = tx + (1-t)y where t = [0,1].

My attempt so far consists of using the definition twice. One for my x1 values 0 and -2. The other attempt is for my x2 values. I would then see if both equation should true but against what value would I be holding them to?

I would like to know where to proceed and/or if there are any issues with my logic.

Thank you.

Answer 1

Your logic is incorrect in many ways, so it's easier to explain what you need to do.

To prove a set $S$ is convex:

1. Take two arbitrary elements $P, Q \in S$.

2. Consider the line segment $tP + (1-t)Q, t \in [0, 1]$.

3. Show that every point on that line segment is in $S$.

In this case $S = [0, 4] \times [-2, 3]$, and so you will have $P = (x_1, y_1), Q = (x_2, y_2)$ where $0 \leq x_1, x_2 \leq 4$ and $-2 \leq y_1, y_2 \leq 3$.

Notice that regardless of the kind of space that $S$ lies in, you always have a single parameter $t$. If you have different parameters for the $x$ and $y$ components, you are no longer drawing a straight line but some kind of surface.