Near Extensions and Alignment of Data in R(superscript)n

7 Tensors, Hyperplanes, Near Reflections, Constants (η,τ,K)

In Chapter 3, we presented a counterexample showing that with , the case  >  in Theorem 2.2 (part(1)) provides a barrier to the existence of the extension there. Moving forward, we now wish to study geometries on the finite set which remove such a barrier. Indeed, we have already made an optimistic guess for the following geometry on the set . Roughly, put: we remove degenerative cases as outlined below.

• The diameter of the set is not too large.
• The points of the set cannot be too close to each other.
• The points of the set are ...