Hi i´m reading a pdf about linear programming and i´m having trouble understanding the difference between a polyhedron and polytope between those two definitions
A polyhedron P ⊆ $R^{n}$ is the set of all points x ∈ $R^{n}$ that satisfy a finite set of linear inequalities. Mathematically, P = {x ∈ $R^{n}$: Ax ≤ b} for some matrix A ∈ R m×n and a vector b ∈ $R^{m}$.
And the definition of polytope is the next one
A polyhedron is called a polytope if it is bounded, i.e., can be enclosed in a ball of finite radius.
So my problem isn´t the definition of polyhedron already bounded , and also are all polyhedron in these context convex?
And for example
{x ∈ $R^{n}$: Ax ≤ b} if i guide myself from the given definitions i can only assure this is a polyhedron but not a polytipe since it isn't technically bounded but is convex and {x|$x\in\mathbb{R}$} isn't any of those two.
Any insight would be helpful