Project an n-simplex of side length $a$ on it's ($n-1$)-dimensional circumsphere by a ray starting at the center. Denote the images of the $n+1$ faces of dimension $n-1$ 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 $d(a_i, a_j)>=a$?
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$\begingroup$ what is your last $a$? $\endgroup$– Gjergji ZaimiCommented Jan 22, 2011 at 19:44
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$\begingroup$ its the side length of the inscribed n-simplex $\endgroup$– bearkillerCommented Jan 22, 2011 at 19:45
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$\begingroup$ I assume "midpoint" is the center of the sphere? $\endgroup$– Igor RivinCommented Jan 22, 2011 at 21:02
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1 Answer
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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 self-evident, 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
answered Jan 22, 2011 at 21:54