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It is:
https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions

I have no idea what that article talks about   But it seems that I have rediscovered it

For a real number p ≥ 1, the p-norm or Lp-norm of x is defined by
= ( abs(x)^power + abs(y)^power + abs(z)^power )^(1/power)

Rounded box DIFS could be something related to it.
https://en.wikipedia.org/wiki/Superegg

The easiest way to understand p-norms (at least for me) is to consider them as different ways to measure distance. For p=2 (the 2-norm), in 2D you get distance = sqrt(x^2 + y^2), which is the Euclidean distance we consider the normal way to measure distance in 2D space. But consider the 1-norm (when p=1): distance = abs(x) + abs(y). The 1-norm is also called the taxicab distance, in reference to a taxicab travelling along an idealized grid of streets. In a taxicab, the important measure (for both driver and passenger) is not the Euclidean distance from point A to point B "as the crow flies", but the taxicab distance which will be the sum of the distances along each street. So p-norms are different but potentially valid ways of measuring distance.

There's been some interesting work recently on applying p-norms to circle and sphere inversions. In a paper by Ramirez et al this is referred to as p-circle inversion, and in a paper by Gdawiec it is referred to as pseudo-inversion. Based on those papers I've done a little bit of work recently with p-norms while developing enhanced circle inversion variations for JWildfire.

In Mandelbulb3D I think ABoxVaryScale2, with the right settings for RPow and XYZPow, essentially uses a p-norm measure for the distance between the iterated point and the origin of the sphere of inversion.

I do like the idea of applying p-norm measurements as options to other formulas. I'll add it in soon as an option for the 2D Mandelbox code I'm developing for Fragmentarium over in the thread "Generalized shape inversion applied to the Mandelbox"

PS Your initial post in this thread motivated me to add a superellipse as one of the shape options for that code

One interesting feature of this is that superfomula is based on this. Superformula is patented.

I just did some thing where each axis had different powers. The only problam I don't know, what to use as mean value for inverse power - root. Geometric mean seems to work best but I 'm not shure is it anywhere correct. Alsou harmonic mean worked nicely but sometimes destroyed far away DE.

its like:
temp = abs(x)^k+abs(y)^l+abs(z)^m
rootpower= 1/ cubic root of ( k*l*m) // geometric mean of power values
radius = temp^rootpower

Of corse these functions are slow.
Not shure how to call this. I called it superseed becouse it looks like seed.


I was thinking that rounded box is something like this. It looks similar but wikipedia sayes it is a different shape. Have no idea about formula. A tought of rounded Abox (forgot the formula). Maybe it was one you mentioned.

Inspired by superformula I made different curvatures of p in each axis. Maybe analogy could be Flat Earth with direction "up" being very different from directions "north" or "east". But realy it was meant to become a core for mandelbulb.box

Mean power of 3 p-s (sCurvatures) is calculated by geometric mean, and then inverted to root:
invpower= 1/ cubic root (sCurvatureX * sCurvatureY * sCurvatureZ)

Optimised:
invpower:=power(PowerX *PowerY *PowerZ*PowerW,-0.33333333333333333333);
(It needs a lot of 3s, this is visible. Numbers of 3s depends on Double or Float)

Then radius is calculated by
R= (abs(x)^sCurvatureX + abs(y)^sCurvatureY + abs(z)^sCurvatureZ)^invpower;

What I got I called a superseed becouse some combinations looks like seeds.
Then it is visialised by iterating radius in modified sigma function:

divisor:= 1/(Sigma+1);
temp:= (Sigma + R )*divisor;
(x,y,z) := temp*(x,y,z)

In Apophysis and in flame fractals there are Fisheye transformation:

radius = square root of( x^2 + y^2)
temp = 2/(r+1)
(x,y) = temp*(x,y)
It generates scenery as fish eye lens but mathematicaly it is related to sigmoid function:
https://en.wikipedia.org/wiki/Sigmoid_function
(Not sigma, may mistake)
I think this is the obvious place were radius can be changed to p-norm.