Polyhedra
Why
We generalize real polyhedra to aribtrary inner product spaces.
Definition
Suppose $X$ is a vector space with an inner
product $\ip{\cdot ,\cdot }$ over $\R $.
A set $P \subset X$ is a
polyhedron (called a
polyhedral set) if there
exists $c_1, \dots , c_m \in X$ and $\alpha _1,
\dots , \alpha _m \in \R $ so that
\[
P = \Set{x \in X}{\ip{x, c_i} \leq \alpha _i \text{ for } i
= 1, \dots , m}
\]
As before, a polyhedron is a polytope if it is bounded.