Definition of extreme point: A point $f$ in $L$ with $\|f\| = 1$ is called an extreme point of the unit ball if $$f = (1 - \alpha)f_1 + \alpha f_2, \, 0 < \alpha < 1, \, \|f_1\|, \|f_2\| \leq 1 \implies f_1 = f_2$$ Let $1 < p < \infty$. Show that every element $f$ of the unit ball of $L_p$ with $\|f\|=1$ is an extreme point.
I have seen proofs of the other direction, but there isn't any for this direction. Any help is appreciated! Thanks
The condition for equality in $\|f+g\|_p\leq \|f\|_p+\|g||_p$ ($f,g \neq 0$) is $f=cg$ with $c \geq 0$. This immediately gives what you want.
Ref. When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?