Geometry of the Banach Spaces

[Valerio Capraro]

Let $X$ a Banach space, $S$ its unit sphere, $X$* the dual of $X$ and $S$*
the unit sphere of $X$*.

Let's give two definitions:

1) $X$ is said to be smooth iff for each $x\inX$ there exist unique $f\in S$* that
attain its norm in $x$.

2) $X$ is said to be strictly convex iff for each $x,y\inS$ then $||(x+y)/2||<1$

Find an example of a smooth Banach space that isn't strictly convex
Find an example of a strictly convex Banach space that isn't smooth.
Prove that if $x,y\in S$ imply $||(x+y)/2||=1$, then the set of $z$ in the line
between $x$ and $y$ such that $||z||=1$ have an accomulation point.