TITLE: A condition that all convex combination of two elements in a uni sphere of Banach space belongs to unit sphere
QUESTION [5 upvotes]: Let $X$ be a Banach space. $S_X=\{x\in X:\Vert x\Vert=1\}$. 
If $x, y, \frac{x+y}{2}\in S_X$, is it true that $\lambda x+(1-\lambda)y\in S_X$ for every $\lambda \in [0,1]$? I can show this if $X$ is a Hilbert space, but in general there is no inner product and no parallel law. Can you give a hint?

REPLY [3 votes]: May assume $$\lambda\in \left(\frac{1}{2},1\right).$$ If $$\Vert \lambda x+(1-\lambda)y\Vert<1,$$ then since $$\frac{1}{2}x+\frac{1}{2}y=\frac{1}{2\lambda}[\lambda x+(1-\lambda)y]+\left(1-\frac{1}{2\lambda}\right)y,$$ $$\Vert\frac{1}{2}x+\frac{1}{2}y\Vert<\frac{1}{2\lambda}\cdot 1+\left(1-\frac{1}{2\lambda}\right)\cdot 1=1,$$ a contradiction.