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Comment by HappMacDonald

> (Aside: Strangely, the relationship is non monotonic and is actually maximal for n=6)

For this aside I crave a citation.

When n=1 the sphere fit is 100% as both simplex and sphere are congruent in that dimension. And dismissing n=0 as degenerate (fit is undefined there I suppose: dividing by zero measure and all that) that (first) dimension should be maximal with a steady decline thereafter thus also monotonic.

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