I am not sure of the notation because you did not tell us about x & y.
But if they are in the unit ball: .
Make the substitutions in the inequality.
If that is not what you need, please explain the difficulty in greater detail.
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:
hence which proves convexity of the ball.
How did they deduce the above inequality?