Proof of the Folding Method

Since G has a fixed point at the origin and G has the centroid of orbit P asAa fixed point (see Lemma 1, Appendix B) we have that w is the centroid of orbit P,  

The points Pv. .., Pm form an orbit of G, thus the following must be satisfied, 

Using Lagrange multipliers with Eqs. 1-3 we must minimize the following, 

Noting that g- are isometries and distance preserving, we have from the derivatives  

The geometric interpretation of Eq. 6 is the folding method as described in Section III, thus proving that the folding method results in the G-symmetric set of points closest to the given set. 

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