In some random excursions to music theory, I came up with a "measure" on a specific vector-like object that is simply defined as the product of all the components, AKA:
$$M(v)=\prod_{k=1}^n v_k$$
Although it is not a norm, I like to think about it as a norm measure of some kind because that intuition works well for the specific music theory application I use it for. Regardless, it has some properties that I can see being desirable in certain contexts:
- There is an intrinsic parity measure that you can opt to eliminate by introducing absolute value signs
- The measure seems to do an exceptional job in measuring the imbalance of spending in limited-budget scenarios. If each component is an asset, each costing the same amount of resources (a weighted version of the norm can generalize that to different costs), and a large sample size of different spending scenarios are visualized pictorially, then the most even split will always have the maximum "norm", and more unbalanced spendings will have lower "norms" that correlate suprisingly well with the visual imbalance of the spending across the different assets. The norm-like usage is inspired by the fact that in aforementioned limited-budget contexts, a simple L2 norm also reaches an extremum at the most even spending, but this time a minimum. Ranking specific spending outcomes in discrete datasets by both the L2 norm and the "product norm" produces a similar (but reverse) ranking, but the "product norm" picks up on the imbalance in spending much much more effectively.
- For datasets of multiplicative nature, a component of 1 doesn't contribute to the score, just like a component of 0 doesn't contribute in Lp norms of vectors in normed vector spaces. Which suggests a norm-like usage for multiplicative datasets (despite not being close to a rigorous norm in any sense).
My question might be a little disappointing, but does anyone know of any usage of such measure, and whether any write-ups about such construct exist? I couldn't find much by searching.