This week, I finished reformulating the 2D case in the dot product notation and using the lagrangian approach. After converting dot products to the equivalent trigonometric representation, we get the same answer as our previous paper. This is exciting. It means that our approach has potential to work.

However, a few road blocks currently exist. I thought about trying to simplify the problem by using a change of coordinates, so that only one angle is relevant to the consensus problem, but this might not be possible as not all the points that we wish to average will exist on the same great circle.

Another idea could be to iteratively apply the algorithm from our last paper, but this could run into a couple problems: Is the formula from the previous paper linear or affine? There also seems to be a "geometric term" that we have been seeing in our lagrangian approach similar to how the DoS in different dimensions is different (analogy from solid-state physics).

However, maybe it is possible to straight shot to the general solution after splitting into even/odd (non-integer bessel functions present in odd dimensions) and obtaining the appropriate ratio of bessel functions to reduce our proof to a "simple" monotonicity proof of the ratio of two bessel functions.

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