I'm trying to understand the norm formed by combining $L^1$ and $L^\infty$ norms as
$||x|| = \max(||x||_\infty, ||x||_1 / \sqrt{d})$
(where $d$ is the dimension of the space). We know that this is a true norm (see Maximum of two norms is norm ) but I'd also like to know if it has other documented properties. For example, how far does it deviate from its component norms (in worst-case, on average), or from another norm such as $L^2$, and how does this vary with dimension?
Is there a general term for norms formed as the combination of other norms? (I'm referring to this as a "composite norm", as I don't know any name for it.)
Is there any other keyword or literature tip you can give, for someone looking to understand the characteristics of this composite norm?
It's hard to search for, since I don't know any other keywords for this concept, and also because there's a lot of content out there that combines norms in other ways: I'm finding it difficult to filter out a lot of irrelevant discussion of adding norms onto cost functions. The literature I'm searching for could simply be explaining basic properties of such a composite norm, or their utility in statistical optimisation, or something else. It could be useful for me to see treatments of similar-ish composites.
