TITLE: Inner product space, points cannot be placed inside a ball of a given radius
QUESTION [9 upvotes]: I've found a very nice problem and I don't know how to go about solving it.

Let $(E, || \cdot ||)$ be an inner product space, $x_1, ..., x_n \in E$.
Prove that if for $i \neq j$ we have $||x_i - x_j|| \ge 2$, then the points $x_1, ..., x_n$ cannot be placed inside a ball of radius less than $\sqrt{\frac{2(n-1)}{n}}$.

$B(x_0, r) = \{y \in E  \ \ | \  \ ||x_0-y||<r\}$
The smallest ball containig two points such that $||x_1 - x_2|| \ge 2$ would have radius $=1$ and $\sqrt{\frac{2 \cdot 1}{2}}=1$ So it works for 2 points.
In order to place three such points inside a ball we need to construct an equilateral triangle, and then the radius $= \frac{2 \sqrt{3}}{3}=\sqrt{\frac{2 \cdot 2}{3}}$. So it works for three points.
However in case of four points we don't get a square but a rhombus (mistake - it's a tetrahedron).
I was hoping I could use induction, but I can't figure out how to estimate the radius of the ball.

REPLY [3 votes]: I think there is a mistake in the example. For $n=4$ the shape is not a Rhombus but a tetrahedron. Imagine $4$ points with equal distance with each other. The resulting geometric shape must be a tetrahedron. Similarly the $n=k$ case should give us a $k$-dimensional simplex. 
The question now is to deduce the ball which can contain a $k$-dimensional simplex with side length $2$. The calculation seems to be already done at here, which I am not going to replicate. This should be enough to solve the problem.