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I'm familiar with various classical means for positive real numbers—such as the arithmetic, geometric, harmonic, and, more generally, the power means.

For points in the Euclidean plane (or in higher-dimensional spaces), the arithmetic mean is straightforward: we simply average the coordinates componentwise. However, if we take the geometric mean of two points componentwise, the resulting mean point does not lie on the line segment connecting the original points. This violates a property I believe should be preserved.

I believe we should define the mean of two points in relation to a third point, ensuring that this definition remains invariant under translation.

I'm curious whether there are standard generalizations of other means—such as the geometric, harmonic, or power means—to points in $\mathbb{R}^2$ (or $\mathbb{R}^n$) that retain desirable geometric properties, like lying on the segment between two points or being symmetric and continuous.

I've searched but haven't found much on this topic, despite it seeming like a natural extension. Any references, papers, or explanations, ideas would be greatly appreciated.

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  • $\begingroup$ A loosely related question here $\endgroup$ Commented May 17 at 9:27

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In the 2D case, you can use the dual view of complex numbers :

  1. Either considered as numbers, allowing to give a meaning to expressions like

$$a=\tfrac12(z_1+z_2), \ \ g=\sqrt{z_1z_2}, \ \ h=\frac{2z_1z_2}{z_1+z_2}$$

((a)rithmetic, (g)eometric, (h)armonic, resp.).

  1. Or considered as points in the so-called "Argand plane", as displayed in fig. 1 where you can see that, for $z_1,z_2$ in the first quadrant, $a,h,g$ are almost aligned, $g$ being almost the midpoint of line segment [a,h].

enter image description here

Fig. 1. The case of 2 points. An example showing a first system of points $(z_1,z_2,a,g,h)$ (on the right) and a second one which is the image of the first one by transformation $z \mapsto (1+i)z$ displaying invariance by rotation and homothety (there is no invariance by translation).

enter image description here

Fig. 2 : The case of 3 points is very similar.

Further investigations can be done :

  1. What is the rationale for the fact that $g$ is almost the midpoint of $a$ and $h$ ?

  2. Generalization to the case of $n$ points.

  3. Looking whether quaternions, which generalize complex numbers to the fourth dimension, could provide a similar treatment (with some doubts due to non-commutativity).

  4. etc.

Matlab program for the generation of Fig. 1

 clear all;close all; z1=rand+i*rand;z2=rand+i*rand; plot([z1,z2],'-ok');hold on a=(z1+z2)/2;plot(a,'or');hold on g=sqrt(z1*z2);plot(g,'og');hold on h=2*z1*z2/(z1+z2);plot(h,'ob');hold on plot([h,a],'-b');hold on L=[z1,z2,a,g,h];d=0.01*ones(1,5) set(gcf,'defaulttextfontsize',15) text(real(L)+d,imag(L)+d,{'z_1','z_2','a','g','h'})
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  • $\begingroup$ Thank you for the insightful visualization. These are great observations, and many of them can be addressed through the properties of power means. It's clearly visible that only the arithmetic mean lies on the segment between the two points. It's also new information for me that, unfortunately, when dealing with three points, these means can fall outside the triangle. I was hoping to find a well-defined notion of means for points, especially since I want to interpret them in the Euclidean plane. For that reason, it's important to me that the definition be invariant under translations. $\endgroup$ Commented May 17 at 11:16

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