convex set and polytope

Question

A polyhedral convex set is defined by

$X=\{x\in R^n:<x,b_i> \leq B_i\} $

My question is :

Is a polyhedral convex set always compact? and what is the difference between a polytope and the set X?

Thank you

Answer 1

The difference is that $X$ may be unbounded (and therefore not compact). For example, if you have only one $b_i$ and $B_i$, then the set $X$ is unbounded (for example, if $B_i\geq 0$, then if some $x$ satisfies the condition $\langle x, b_1\rangle =0$, then $\lambda x$ also satisfies the same condition.