All regular simplexes are congruent

Question

In n dimensional Euclidean space, I read that the definition of the regular simplex is the convex hull of n+1 points such that: (i) the distance from any of the points to their centroid is constant. (ii) the distance from any two of these points is constant.

A few things were asked about said definition, and I’m struggling on pretty much all of them. In particular, how can I show that these points are in general position? And how can I show that any two regular simplexes are congruent (so such that there exists an isometry which maps one to the other)?

Thanks in advance

Edit: another tough question asked to calculate the distance between any of the points and the centroid as a function of the distance between any two of the given points.

Answer 1

I am unsure whether part (i) of the definition is even required to define a regular simplex, part (ii) should suffice. It all follows from the following result.

Lemma: Let $f: S \subset \mathbb{R}^n \rightarrow \mathbb{R}^n$ be an isometry, then $f$ may be extended to an isometry $\tilde{f} : \mathbb{R}^n \rightarrow \mathbb{R}^n$.

Proving the lemma is mainly just fiddly as we have to check the cases where the affine span of $S$ is not the whole of $\mathbb{R}^n$. The easiest way to prove it is under the assumptions that $0 \in S$, $f(0)=0$ and span$(S) = \mathbb{R}^n$, then we apply that preserving distance is equivalent to preserving the induced dot product.

Now suppose we have two regular simplices $S:= \{s_0, \ldots,s_n\}$ and $S':= \{s'_0, \ldots,s'_n\}$ of $n+1$ points such that $\|x-y\|=1$ for all distinct $x,y \in S$ and for all distinct $x,y \in S'$. If we let $f : S \rightarrow S'$ be the map $f(s_i)=s'_i$ then it is an isometry, thus by our lemma we can extend this map to an isometry $\tilde{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $S,S'$ are congruent. This means that all regular simplices of the same size are congruent to each other.

If we supposed that $S$ was in general position but $S'$ was not, then the isometry $\tilde{f}$ would not be injective, as it will map the span of $S$ into the "flat" span of $S'$, contradicting that isometries are injective. Since any regular simplex has a placement in general position then this proves all regular simplices are in general position.