A criterion for determining co-sets in finite dimensional real vector spaces

Question

Let $V$ be a finite dimensional vector space over $\mathbb R$.

Let $A\subseteq V$ such that for every finite set $a_1,...,a_n \in \mathbb R$, with $\sum_{i=1}^na_i=1$, and every $v_1,...,v_n\in A$ , $\sum_{i=1}^na_iv_i \in A$ . Then how to show that $A$ is a co-set in $V$ i.e. $A=x_0+W$ for some $x_0 \in V$ and linear subspace $W$ of $V$ ?

Answer 1

Step 1 : Say $0\in A$. Then for any $v,w\in A$ and $c\in\mathbb{R}$ you have $c\cdot v = c\cdot v + (1-c)\cdot 0\in A$ and $v+w=v+w-0\in A$ so $A$ is a subspace.

Step 2 : If $0\not\in A$ take any $x_0\in A$, and define $W$ as $A-x_0$ as in the comments. Use step 1 (note that $A-x_0$ satisfies the same conditions) to conclude that $W$ is a subspace.