Suppose $x$ is an extreme point of the unit ball of a Banach space $E$. Embed $E$ in $E^{\*\*}$ in the standard way. Is $x$ an extreme point of the unit ball of $E^{\*\*}$?



Weak* extreme points
The answer is **no** and $E=\mathscr{K}(\ell\_p)$ for $p\in (1,\infty)\setminus \{2\}$ is already a counterexample. See the proof of Proposition 2.3 in



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> J. Hennefeld, [Compact extremal operators](http://projecteuclid.org/euclid.ijm/1256049500), *Illinois J. Math.* **21** (1997) 61-65.
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Here we identifty $\mathscr{K}(\ell\_p)^{\*\*}$ with $\mathscr{B}(\ell\_p)$.


This question is discussed in:



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> S. Dutta, T. S. S. R. K. Rao, [On Weak-${}^\ast$ Extreme Points in Banach Spaces](http://home.iitk.ac.in/~sudipta/revise.pdf), *Journal of Convex Analysis*, **10** (2003), No. 2, 531-539.
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where further examples are given.


