# convex polytopes – Intrinsic definition of a cone in a normal fan

Let $$Psubseteq mathbb{R}^n$$ be a full dimensional polytope. Let us assume that $$P$$ has a facet description with the following inequalities:
$$left geq -a_F$$
where $$u_Fin mathbb{R}^n$$ and $$a_Fin mathbb{R}$$ for each facet $$F$$ of the polytope.

In other words, $$P$$ is the bounded intersection of the halfspaces $$H_{u_F,-a_F}^+$$. For each $$Q$$ face of $$P$$ we define the cone
$$sigma_Q := operatorname{Cone}(u_F: F text{ facet, } Fsupseteq Q)$$

What I want is to prove that $$sigma_Q = {uin mathbb{R}^n : left leq left text{ for all } xin Q, yin P}$$

My thoughts: It is easy to prove one of the inclusions. For the other, I wanted to use (one of the many versions of) Farkas Lemma.

If $$uin mathbb{R}^n$$ satisfies $$leftleq left$$ for each $$xin Q$$ and $$yin P$$, in particular, there exists a face $$Q’$$ of $$P$$ such that $$Q’ = P cap H_{u,a}$$ for certain $$a=left$$ for any $$xin Q$$ (this value is independent of $$xin Q$$), also $$Qsubseteq Q’$$ and $$Psubseteq H_{u,a}^+$$. I tried to use Farkas Lemma II (see Ziegler’s book). So, to conclude that $$uin sigma_Q$$, I need to prove that there cannot exist a vector $$min mathbb{R}^n$$ with the property that:
$$left geq 0 text{ for all facets } Fsupseteq Q’$$
and simultaneously
$$left < 0.$$

To get a contradiction I think it suffices to show that one can find a point $$xin Psmallsetminus Q’$$ such that $$x = q – lambda m$$ for a $$qin Q’$$ and $$lambda > 0$$.

In that case, $$xin Psmallsetminus Q’$$ implies that for some facet $$F$$ with $$Fsupseteq Q’$$ it has to be: $$left > -a_F$$, which yields $$left – lambdaleft > -a_F$$. The fact that $$qin Q’$$ implies that $$left = -a_F$$, from where it follows $$lambdaleft < 0$$ and $$lambda > 0$$ yields $$left < 0$$ which is the desired contradiction. I am having trouble to prove the existence of such an $$x$$.

Of course, a different approach to prove this is also welcome!