Q:

Is $2n$ the smallest number of halfspaces to determine a segment in $\mathbb{R}^n$?

I proved that a segment in $\mathbb{R}^n$ is a polyhedron, and it is determined by $2n$ halfspaces. My question follows: 

Is $2n$ the smallest number of halfspaces to determine a segment in $\mathbb{R}^n$?

I assumed the contradiction but still got nothing. I think it may be related to some properties of adjacent extreme points.
Any help will be appreciated. Thank you very much.

A:

Take the line segment in $\mathbb{R}^3$ with endpoints $(0,0,0)$ and $(0,0,1)$.  It is defined by these five halfspaces:
$$x+y\leq0,y\geq0,y-x\leq0,z\geq0,z\leq1$$

