# Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.

If $P$ is our point, then it will be contained in the ball corresponding to edge $(i,j)$ if and only if the angle $\angle iPj$ is greater than or equal to $\frac{\pi}{2}$. If there is no such edge $(i,j)$, then every vertex $j$ is on a fixed side of the hyperplane through $P$ orthogonal to the line connecting vertex $1$ with $P$, so $P$ is outside the simplex.
• Can you please expand on the logic behind the last "so..."? Every vertex $j \neq 1$ is to one side of that hyperplane. How does this imply $p$ is outside? – Joseph O'Rourke Aug 22 '13 at 12:37