Let $S_1$ and $S_2$ be two regular $n$-simplices (in $\mathbb{R}^{n}$) with edge length $\sqrt{2}$. Furthermore assume that the vertices of $S_1$ and $S_2$ are integer points (that is having each co-ordinate as integers). I am trying to prove the intersection of $S_1$ and $S_2$ is a $k$-simplex for some $k\leq n$, and $k=n$ if and only if all the vertices of $S_1$ and $S_2$ coincide (i.e. $S_1=S_2$).
I tried in $2$ and $3$ dimensions, and it seems like this should be true in higher dimensions. But I am not able to write a rigorous proof of this.
I tried a google search and the best thing that I could find is this.
I tried to show that any regular $n$ simplex with integer vertices is uniquely determined by it's centroid. I am not sure how to prove this and whether this is sufficient to prove what I originally want.
Any help or reference will be greatly appreciated.
:)
There are a lot of questions here, but in $\Bbb R^3$ consider the simplex $A$ with vertices $(0,0,0)$, $(1,1,0)$, $(1,0,1)$ and $(0,1,1)$, and the simplex $B$ with vertices $(0,0,1)$, $(1,1,1)$, $(1,0,0)$ and $(0,1,0)$.
Both $A$ and $B$ are regular with side-length $\sqrt2$, they have the same centre and their intersection is not a simplex.