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Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, given by

||A||p + ||B||p = ||C||p

where A, B and C are still non-negative, but we relax normalization on A and B. Imagine that C is fixed and, without loss of generality, normalized. We want to solve for A and B.

First, note that one obvious family of solutions is

A = (1-x) C , B = x C , 0≤x≤1 .

Question: Ignoring the obvious permutation symmetries, are these the only solutions?

Edit: By p-norm, I mean the vector p-norm: ||A||p = (∑j |aj|p )1/p. Although we don't really need the absolute values, since the aj are all non-negative.
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Are two probability distributions uniquely constrained by the sum of their p-norms?

Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, given by

||A||p + ||B||p = ||C||p

where A, B and C are still non-negative, but we relax normalization on A and B. Imagine that C is fixed and, without loss of generality, normalized. We want to solve for A and B.

First, note that one obvious family of solutions is

A = (1-x) C , B = x C , 0≤x≤1 .

Question: Ignoring the obvious permutation symmetries, are these the only solutions?