an $L^p$ implication

Question

Let $1<p<\infty$ and $x,y\in L^p([0,1])$ such that $\|x\|_p = \|y\|_p = 1$. Then the following implication holds:

$$\left\|\frac{x+y}{2}\right\|_p=1\Rightarrow x=y\tag{*}$$

This does not hold for $L^1([0,1])$.

I tried proving this, but I couldn't get anywhere. My idea is to look at $\|x-y\|_p$ and show that this norm is $\leq 0$ and hence $x=y$ due to the positivity of norms. I tried adding $\pm\frac{x+y}{2}$ to use the triangle equality and (*), but this did not get me anywhere.

How can I prove this implication?

Answer 1

Hint: Use Clarkson's inequalities