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If $$\sum_i \lambda_i = 1 = \sum_j \mu_j$$ are finite sums, then $$\sum_i \lambda_i a_i + \sum_j \mu_j b_j = \sum_{ij} \lambda_i \mu_j (a_i + b_j).$$ Is this formula just a formula or can you give some nice interpretation as the homomorphism property of some map between appropriate algebraic structures with the operations like an alternate multiplication or addition or something along these lines?

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    $\begingroup$ Every formula is more than just a formula if you chew on it for long enough. ;-) $\endgroup$ Jul 21 at 18:58

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If we divide through by 2, we get $$\frac12 (\lambda^i a_i + \mu^j b_j) = \lambda^i\mu^j \frac{a_i + b_j}2.$$ In words, the midpoint of {a convex combination of the $a_i$} and {a convex combination of the $b_j$} is itself a convex combination of the midpoints $(a_i + b_j)/2$.

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