Commit 2020-06-30 03:05 791744b0

feat(analysis/normed_space/real_inner_product,geometry/euclidean): inner products of weighted subtractions (#3203) Express the inner product of two weighted sums, where the weights in each sum add to 0, in terms of the norms of pairwise differences. Thus, express inner products for vectors expressed in terms of weighted_vsub and distances for points expressed in terms of affine_combination. This is a general form of the standard formula for a distance between points expressed in terms of barycentric coordinates: if the difference between the normalized barycentric coordinates (with respect to triangle ABC) for two points is (x, y, z) then the squared distance between them is -(a^2 yz + b^2 zx + c^2 xy). Although some of the lemmas are given with the two vectors expressed as sums over different indexed families of vectors or points (since nothing in the statement or proof depends on them being the same), I expect almost all uses will be in cases where the two indexed families are the same and only the weights vary.