Let $C$ be a convex subset of a vector space $X$. A point $x\in C$ is called an extreme point if and only if whenever $x=ty+(1−t)z$, $t\in (0,1)$, implies $x=y=z$. It is known that the boundary points in the closed unit ball are extreme. But this will happen at all border points when this is "curve"? As one might define this notion of "curve", how to characterize the convex set whose boundary is formed by points extrem?
There are many questions, I hope to clarify the picture. Thanks for your time and patience.