Full-dimensional convex subset of $\mathbb{R}^n_{++}$

Question

Suppose $A\subset\mathbb{R}^n_{++}$ is convex and $n>2$.

Suppose also that for all $i=1,\ldots,n$ there exist $x,y\in A$ and $j\neq i$ such that $x_i>y_i$ and $y_j>x_j$.

Question: Does $A$ contain $n$ linearly independent elements?

Answer 1

If I get it right, this $A$ does not contain $n$ linearly independent points. Let us consider $n=3$, and let $A$ be the segment connecting the points $x=(1,1,0)$ and $y=(0,0,1)$. If $i=1$ or $2$, then let $j=3$, and let $x$ and $y$ be the ones above. If $i=3$, let $j=1$ and switch $x$ and $y$.