The study of norms is generally limited to those that are most interesting, most notably the Euclidean $2$-norm (and rarely the Manhattan $1$-norm, and supremum $\infty$-norm).
However, I can distinguish two important generalizations of norms.
Pseudonorms (ie, euclidean norms where a given coordinate is squared then multiplied by $-1$, before being summed, like the norm for split-complex numbers, Minkowski spacetime, etc). To my knowledge the systematic study of these (quadratic forms), and their geometric consequences, is Clifford algebra.
$\lambda$-norms for $\lambda \in [1, \infty]$. To my knowledge, it's mostly measure theory and functional analysis that deal with these.
Here are my questions, which are subquestions of "how does one generalize the tools from Euclidean (or pseudo-Euclidean) geometry to all norms ?":
Why do we not study $\lambda$-norms for $\lambda \in \Bbb R^* \cup \{-\infty, +\infty\}$, since the formula $(\sum_{i=1}^{i=n} |x_i|^\lambda)^\frac{1}{\lambda}$ is easily generalizable to more values of $\lambda$, and seems to give coherent results (for everything except $\lambda = 0$) ? We barely study anything else than $\lambda = 1, 2, \infty$. Sure, these resolve nicely. But is the reason for this preference simply because it's too complex to compute the other norms ? Or is there some important critical property that fails for these other norms ? I notice that there are nonzero null-norm elements for the negative $\lambda$ (to be precise, the component axes are all undefined for a negative $\lambda$-pseudonorm, but converge to zero continuously, though not smoothly); but this isn't enough do deter from the study of (2-ic, quadratic) pseudonorms, it seems.
Is there a way to cleanly define a $0$-norm via a limit ? (My gut and quick experiments say "probably not".)
Is there any study of $\lambda$-norms where $\lambda$ can take complex values ?
Does there exist any systematic study of $\lambda$-pseudonorms ? By this I mean, for example, the study of a Manhattan norm in 2d, but where the $y$ component is weighed negatively, ie $\|p\| = x - y$. Admittedly, you'd need to consider all possible roots of $x^\frac{1}{\lambda}$ to retrieve negative values of the pseudonorm (and generalize hyperbolic geometry to non-Euclidean pseudonorms), so this is complicated. (Normal quadratic forms just do away with the fractional power to ignore this problem.)
Do concepts originating in multivariable calculus differ when defined with a non-euclidean norm (a fortiori, a non-euclidean topology) ? I am aware of a couple of things like absolute convergence, etc, which are clearly based on non-euclidean norms. But if someone can draw a more complete picture of the path from a $\lambda$-norm to $\lambda$-continuity, $\lambda$-derivability (if these even mean anything...).
What are the tools that can be appropriately generalized to $\lambda$-pseudonorms ? For example, elliptic trigonometry can be extended to hyperbolic trigonometry in the split-complex plane with the "hyperbolic $|1|$-ball" (set of points such that $\|p\| = 1 \text{ or } \|p\| = -1$ for a given pseudonorm $\| \cdot \|$), and the $\cosh$, $\sinh$ etc functions. Also, a quick thought experiment leads me to believe that the equivalent of a "sine wave" for the $1$-norm would be a triangle wave (which to my knowledge is non-negilibly important in sound engineering and signal theory). Is there a way to generalize properties of the exponential and trigonometric functions (formulae, relationships, expansions, role in differential equations, ...) to their version for arbitrary $\lambda$-pseudonorms ?
I'm aware this might be a pretty open question, but since it's a lack of mathematical culture on my part, perhaps there's some neat theory with a name I don't know that deals with this, and you can just recommend a textbook ! I thank you for your time reading and answering this in any case. (You are welcome to edit the tags if you find them inappropriate.)
Finally some pictures. We plot the various curves (in 2D) of norm $1$ (or $-1$):
- with $\lambda$ (as $n$) taking the values $5$, $2$, $1$, $0.5$, $-0.5$, $-1$, $-2$, and $-5$.
- with both the "complex" "lambda-ic" form ($x$ and $y$ same sign in the form) and the "split-complex" "lambda-ic" form ($x$ and $y$ opposite signs in the form).
