Face of a simplex, convex hull

[Linux]

For a point $p$ in a simplex we define $ \text{supp (p)}= \{ i \in I_n \ | \ p_i \neq 0\}$

For a convex set $C$ we define $F$ to be its face if $F$ is convex and $\forall x,y \in C, \lambda \in (0,1) : \lambda x + (1- \lambda ) y \in F \Rightarrow x,y \in F$

For $A \subset \{1,2,...,n \}$ we define $F(A) : = \{ p \in \Delta_n \ | \ \text{supp(p)} \subset A \}$

Could you tell me how to prove that $$\text{conv} (F(A_1) \cup F(A_2)) = F(A_1 \cup A_2)$$

Here this inclusion is quite evident "$\subset$" and I've proved it using $2)$ below and the fact that for $A_1 \subset A_2$ we have $F(A_1) \subset F(A_2)$ (precisely the fact that both sets are contained in their union)

I know that:

$1) F \text{ is a face } \iff \exists A \subset I_n : F = F(A)$

$2) \text{conv} (\{ e_i, i \in A \} ) = F(A)$ (canonical basis of $\mathbb{R}^n$)

$3) F(A_1 \cup A_2) = F(A_1) \cup F(A_2)$ if $A_1 \subset A_2$ or $A_2 \subset A_1$

But I don't know how and if to put those facts together.

Could you help me a bit?

 

[jvicens]

Sure, I'd be happy to help you with this proof. First, let's review the definitions that we have been given:

1. For a point $p$ in a simplex, $\text{supp}(p)$ is the set of indices $i$ such that $p_i \neq 0$.

2. For a convex set $C$, a face $F$ is a subset of $C$ that is itself convex, and for any two points $x,y \in C$ and any $\lambda \in (0,1)$, if $\lambda x + (1-\lambda)y \in F$, then $x,y \in F$.

3. For a subset $A \subset \{1,2,...,n\}$, $F(A)$ is the set of all points $p$ in the simplex such that $\text{supp}(p)$ is a subset of $A$.

Now, let's break down the proof into two parts: the inclusion $\subset$ and the inclusion $\supset$.

First, let's prove that $\text{conv}(F(A_1) \cup F(A_2)) \subset F(A_1 \cup A_2)$. This means we need to show that any point $p$ in the convex hull of $F(A_1)$ and $F(A_2)$ is also in $F(A_1 \cup A_2)$.

To do this, let $p$ be a point in the convex hull of $F(A_1)$ and $F(A_2)$. This means that $p$ can be written as a convex combination of points in $F(A_1)$ and $F(A_2)$, i.e. there exist points $p_1 \in F(A_1)$ and $p_2 \in F(A_2)$ and scalars $\lambda_1, \lambda_2 \in [0,1]$ such that $p = \lambda_1 p_1 + \lambda_2 p_2$ and $\lambda_1 + \lambda_2 = 1$.

Since $p_1 \in F(A_1)$, we know that $\text{supp}(p_1)$ is a subset of $A_1$. Similarly, since $p_2 \in F(A_2)$, we know that $\text