# Number of faces in a simplex

Let $D \subset \mathbb R^d$ be a $d$-simplex with vertices $V=\{ v_1,v_2, \ldots, v_{d+1} \}$

Prove that for every $W \subset V$ $\operatorname{conv}(W)$ is a face of $D$

My definition of a face of a polytope is the following:

Let $H$ be a supporting hyperplane of the polytope $P$, the faces of $P$ are the sets $P \bigcap H$

A simplex is a $d$-polytope with $d+1$ faces of dimension zero (vertices).

How can I use this to proof the statement above ? I am just learning about polytopes and have no clue how to proceed

• I think you mean "vertices" where you write "edges". The $3$-simplex is a tetrahedron in $3$-space, with $3+1 = 4$ vertices and $6$ edges. Possible hint: an affine transformation can move the vertices to the origin and the endpoints of the standard unit coordinate vectors. Jun 1, 2018 at 21:01
• @EthanBolker Y, I was mixing up the definition of faces with dimension 1 and 0. My fault. So you mean I could "simpy" look at the standard-simplex using the affine transformation ? Jun 1, 2018 at 21:04
• I think it is indeed sufficient to prove this for the standard simplex. Jun 2, 2018 at 0:04

So we have that $$D = \operatorname{conv}(e_1, e_2, ..., e_{d+1})$$. Now let $$W \subset \{e_1, ..., e_{d+1} \} = V$$. Let $$c = \sum_{e_i \in V \setminus W} e_i$$. For example, if $$W = \{e_1, e_2, ..., e_{d-1} \}$$, then $$c = e_d + e_{d+1}$$.
Consider the hyperplane $$H = \{ x \in \mathbb{R}^{d+1} | \langle c,x \rangle = 0 \}$$. Notice that $$\langle c, e_i \rangle$$ is $$0$$ if $$e_i$$ is in $$W$$, and $$1$$ if $$e_i$$ is not in $$W$$.
It follows quickly that that $$D \cap H = \operatorname{conv}(W)$$. Can you show this? Hint: Take $$x \in D$$, write $$x = \sum_{i = 1}^{d+1} \lambda_i e_i$$ s.t. $$\sum \lambda_i = 1$$, $$\lambda_i \geq 0$$, when is $$\langle c, x \rangle = 0$$ ?
The geometric intuition behind the last step is that all in $$V \setminus W$$ lie on the positive side of the hyperplane, while the vectors in $$W$$ lie inside the hyperplane. So in order to stay inside the hyperplane, you can only take the directions offered to you by the vectors in $$W$$.