# Math Help - Convexity of Unit ball

1. ## Convexity of Unit ball

Am I right in saying that the unit ball is convex if for any 2 points in the unit ball, the straight line segment between them is also in the unit ball?

The book uses Cauchy-Schwarz to prove convexity of the ball. Then they do the following:

$s^{2}|x|^2+ 2st|x||y| + t^{2}|y|^2 \leq (s+t)^2 = 1$ hence $|z | \leq 1$ which proves convexity of the ball.

How did they deduce the above inequality?

$z = sx+ty$ where $0 \leq s,t \leq 1$.

2. I am not sure of the notation because you did not tell us about x & y.
But if they are in the unit ball: $\left| x \right| < 1,\;\left| y \right| < 1,\;\left| x \right|^2 < 1,\;\left| y \right|^2 < 1$.
Make the substitutions in the inequality.

If that is not what you need, please explain the difficulty in greater detail.

3. Oh ok, I get it now.

Thanks