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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.

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Can you clarify these two sentences: "But this will happen around that borders convex "curve"? As one might define this notion of "curve", how to characterize the convex sets having point extrem across border point?" –  littleO Nov 15 '12 at 12:55
    
Your definition of an extreme point is wrong as it stands. An extreme point of a convex set $C$ is a point $x \in C$ that cannot be written $x = ty + (1-t)z$ with $t \in (0,1)$, $y,z \in C$ and $z \neq y$. –  Martin Wanvik Nov 15 '12 at 13:02
    
sorry .. try to be more careful –  helmonio Nov 15 '12 at 13:08
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