# Do a set of vertices uniquely determine a polytope?

The question in title arises because I am trying to prove that a polytope is the convex hull of its vertices, i.e., $$\mathcal{P}=conv(V)$$. Here is how far I have got.

Convex hull of a finite set of vectors is a polytope. So for $$v_1,...,v_k$$, $$\mathcal{Q}=conv(v_1,...v_k)$$ is a polytope and I can show that for this polytope, $$\mathcal{Q}$$, $$v_i$$ must be vertices for all $$i$$. So I now know that if I take convex hull of a set of vectors, I get a polytope whose vertices are those vectors. But I require that vertices determine a polytope uniquely or else I cannot complete my proof. Is this true and if not, how do I prove that a polytope is convex hull of its vertices?

• Are you only interested in polytopes that are known to be convex? Commented Aug 9, 2020 at 9:31
• @ChrisCulter Yes, I am only interested in convex polytopes. Commented Aug 9, 2020 at 11:55

Yes, in general this is how one defines a convex polytope. In general, a convex polytope is defined as the convex hull of a finite set (of vertices), page 14:

Def: A convex polytope $$P\subset \mathbb{R}^n$$, or symply a polytope, is defined as the convex hull of a non empty finite set $$\{x_1,\dots,x_q\}$$

So maybe you should provide the definition of convex polytopes that you are using. There another way to characterize a convex polytope $$P$$. Which is by means of a finite set of inequalities. Some times this is refered as the H-V theorem were H stands for half-spaces and V for vertices. Sometimes also as the Fundamental Theorem of Polytopes:

Theorem 9.2: A non-empty set $$P$$ of $$\mathbb{R}^n$$ is a (convex) polytope if, and only if, it is a bounded polyhedral set.

Polyhedral set is defined in the same text as

Def: A subset $$Q$$ of $$\mathbb{R}^n$$ is called a polyhedral set if $$Q$$ is the intersection of a finite number of closed halfspaces, or $$Q=\mathbb{R}^n$$.

• Thank you for your answer. The definition of (convex) polytope I am using is precisely the second one that you mention, i.e., as a bounded polyhedral set. This is why I think I need to prove that convex hull of vertices of a polytope is indeed the polytope itself. I just realised that in the sketch of proof I gave in my question description, I inadvertently used another definition (the first one you mention) as well, so that seems wrong now. How do I go about proving this? Commented Aug 11, 2020 at 9:27
• The proof is kind of intricate and complicated. In the literature I have provided there is a proof, but if you would like more of a sketch you can see this set of videos here, more specifically Lecture 3.
– R.W
Commented Aug 11, 2020 at 13:00
• Thank you very much, I will have a look. Commented Aug 11, 2020 at 16:32