Let $B_1(X)$ be the closed unit ball of Banach space $X$. Let $x\in X\setminus B_1(X)$ and $x_1=\dfrac{x}{\|x\|}$.
Question: Is it true that for all $y\in B_1(X)$, $\|x-y\|\ge\|x_1-y\|$?
If it is not true, can you give me a counterexample? And what about strictly convex space or uniformly convex space?
Not in general, for instance see the example below in $\mathbb{R}^2$ with the $l^{\infty}$ norm.
One can modify slightly the ball to make the space strictly convex, (so uniformly convex, being finite dimensional), for instance $\mathbb{R}^2$ with $l^p$ norm for $p$ large enough. This does not happen in euclidian $\mathbb{R}^2$. To see this geometrically:the tangent from the center of a disk to a second disk with center in the first one touches the second one inside the first disk. Indeed, the lenght of the tangent is at most the distance between the centers.