How to describe the unit ball as an Intersection of hyperplanes? How can one describe the unit ball in $\mathbb{R}^{3}$ as an intersection of supporting halfspaces?

Let $B_3\subset \mathbb{R}^3$ denote the unit ball. By supporting halfspace, I assume you mean an affine half space containing $B_3$ such that its boundary contains at least one boundary point of $B_3$. Taking
$$B_3=\bigcap_{|x|=1}\{x+z:\langle z,x\rangle\leq 0\}$$
should work. Here is a good picture of the lower dimensional case; can you visualize what this looks like in $\mathbb{R}^3$?
