Line segments lying on the unit sphere

Suppose $V$ is an inner product space and $B$ is the open unit ball in $V$ (thus $B = \left\{ f \in V:\left\| f\right\| < 1\right\}$). Prove that if $U$ is a subset of $V$ such that $B \subseteq U \subseteq \overline{B}$, then $U$ is convex.

Give an example to show that the result in part (a) can fail if the phrase inner product space is replaced by Banach space.