Project an nsimplex of side length $a$ on it's ($n1$)dimensional circumsphere by a ray starting at the center. Denote the images of the $n+1$ faces of dimension $n1$ of the simplex by $A_1,\dots A_{n+1}$. Choose $a_i\in A_i, i=1,\dots, n+1$. Are there always $i, j\in \mathbb{N}$ with $d(a_i, a_j)>=a$?

$\begingroup$ what is your last $a$? $\endgroup$ – Gjergji Zaimi Jan 22 '11 at 19:44

$\begingroup$ its the side length of the inscribed nsimplex $\endgroup$ – bearkiller Jan 22 '11 at 19:45

$\begingroup$ I assume "midpoint" is the center of the sphere? $\endgroup$ – Igor Rivin Jan 22 '11 at 21:02
NO
For a regular tetrahedron in $\mathbb E^3$, if you take a centroid of one face together with the midpoints of its three edges projected to the circumscribed sphere, they satisfy your conditions (or if you want the conditions to be strict, each midpoint of an edge can be perturbed into the adjoining face). The distances are shorter than the edgelengths of the tetrahedron. This is geometrically selfevident, but if you want numbers, for the unit sphere, the edgelengths of the inscribed regular tetrahedron are $1.63299...$, and the edgelengths for the tetrahedron of the $a_i$ described above are $1.41421...$ and $.919402...$.
alt text http://dl.dropbox.com/u/5390048/TetrahedronPoints.jpg