I want to prove the following statement:
If the unit sphere in a normed space contains a segment, then there exists two vectors $x,y$ such that $\Vert x + y \Vert = \Vert x \Vert + \Vert y \Vert$ with $x$ and $y$ linearly independent.
This is what I have done: Let $x$ and $y$ be the end point of the segment contained in the unit sphere. Then $\forall \lambda \in (0,1)$ I can write:
$$ \Vert \lambda x + (1-\lambda)y \Vert \leq 1 $$ From here I am struggling on getting the relationship on $x+y$. Any pointers?
Let $x$ and $y$ the endpoints, as you write. Therefore we have $$ \|\lambda x + (1-\lambda)y\| = 1 $$ for all $\lambda \in (0,1)$. For $\lambda = \frac 12$, we have $$ \frac 12\|x+y\| = 1 \iff \|x+y\| = 2 = \|x\| + \|y\|. $$ If $x$ and $y$ were dependent, as both are element of the unit sphere, we would have $x = \mu y$ for some $|\mu| = 1$. Then $$ |\lambda \mu + (1-\lambda)| = 1 $$ for all $\lambda \in (0,1)$, which is impossible for $\mu \ne 1$.