Why do we care about Quasi-norm?

Question

I have understood that quasi-norm does not satisfy the inequality triangle.

I also know that the norm is a function on vector space that assigns length or size to the vector.

However, I do not exactly understand quasi-norm.

I do not really understand why we want to study them within statistics and machine learning.

I was wondering if anyone knew a nice intuitive motivation for the study of quasi-norm?

Thanks in advanced

Answer 1

There are true statements where one doesn't need the triangle inequality. People wondered which are the minimal requirements to prove these statements, and likely that way quasinorms emerged, like much of mathematics.

True, an undergraduate may not need it, or maybe not even most mathematicians, but it has its uses. Arxiv alone lists 3 entries for the query "quasinorm" and 36 entries for "quasi-norm". Note that these may not be all synonyms of that notion.