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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

user643287
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  • You can use Clarkson's inequality to show that these spaces are strictly convex. Any point in the sphere of a strictly convex space is an extreme point of the ball – user124910 Apr 07 '19 at 21:39

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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 $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?

Kavi Rama Murthy
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