Let $B\_1, B\_2$ be unit balls in finite-dimensional normed spaces $X\_1, X\_2$ respectively.
Let $e(B\_1), e(B\_2)$ be corresponding extreme points sets.


Consider the unit ball $B$ in tensor product $X\_1\otimes X\_2$ with the largest (projective) cross-norm on it. 



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> Can we say that extreme points of $B$ in tensor product are exactly tensor products of extreme points for $B\_1, B\_2$, i.e. $e(B)=\{u\otimes w: u\in e(B\_1), w\in e(B\_2)\}$? 
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This seems plausible, but things are not looking very straightforward. In particular, opposit pairs of extreme points produce the same point in tensor product, i.e. $(-u)\otimes (-w) = u\otimes w$. 



Extreme points of unit ball in tensor product of spaces
See


[11] Ruess, W.M. and Stegall, C.P., Extreme points in duals of operator spaces, Math. Ann., 261 (1982), 535–546. 


They prove what you want in a more general context: If $X$, $Y$ are Banach spaces s.t. either $X^\*$ or $Y^\*$ has the approximation property and either $X^\*$ or $Y^\*$ has the Radon-Nikodyn property, then the extreme points of the unit ball of the projective tensor product of $X^\*$ and $Y^\*$ are the tensor products of extreme points of the respective unit balls.


