Distance between Two points can be expressed as

$(|x_1 - x_2|^m + |y_1 - y_2|^m)^{\frac{1}{m}}$

$m = 1$ 1 norm distance (Manhattan Distance)

enter image description here

$m = 2$ 2 norm distance (Euclidean Distance)

enter image description here

How can I visualize 3 norm, 4 norm,... distance. I searched internet and all I could find is the comparison of 1 norm and 2 norm. I couldn't be able to find the visualization of 3 norm, 4 norm, ... distance. Can anybody help me to visualizes these distances.


Here's a picture for you: $$\color{blue}{‖x‖_1 = 1} \quad ‖x‖_2 = 1 \quad \color{orange}{‖x‖_5 = 1} \quad \color{red}{‖x‖_∞ = 1}$$ enter image description here

(It is sufficient to consider the unit balls centered at 0 because it "doesn't matter" where the centre is.) To understand this diagram, pick any point on the corresponding ball; that point is of $p$-distance $1$ from $0$.

As far as I know, the $p$-norms for $p=1,2,∞$ are the only cases that have a clear interpretation. Everything else is an 'interpolation' in between; if $p≈ 1$, then its the $1$-norm, but fudged slightly. If $p\gg 1$ then its like the $∞$-norm. As you can see, $p=5$ is already pretty close to the $∞$-norm.

  • $\begingroup$ $||x||_2$ is Euclidean Distance. So I think It would be Blue figure rather Black figure. $\endgroup$ – Atinesh Oct 22 '15 at 13:31
  • $\begingroup$ @Atinesh well the black one is a circle, which should be the 2 norm. The blue one is a diamond; thats the 1-norm ball. $\endgroup$ – Calvin Khor Oct 22 '15 at 13:32
  • $\begingroup$ I'm confused as far as I know Euclidean distance can be calculated using Pythagoras theorem. How can we apply that theorem in case of Curve. $\endgroup$ – Atinesh Oct 22 '15 at 13:35
  • $\begingroup$ @Atinesh Sorry, but what do you mean? Euclidean distance is defined to be $‖x-y‖_2 := \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2}$. This is a definition, not a theorem. $\endgroup$ – Calvin Khor Oct 22 '15 at 13:38
  • $\begingroup$ Please see this s27.postimg.org/ufsb392cz/Untitled.png $\endgroup$ – Atinesh Oct 22 '15 at 13:57

You could try plotting the unit sphere of these norms, i.e. the set of points at an $m$-distance 1 from the origin. Plot the curve for several different values of $m$ between 1 and infinity. You will see that the unit balls are nested within each other. The diamond of the $m=1$ ball "morphs" into the familiar Euclidean ball, and out the other side, where it morphs into the $m = \infty$ ball. These shapes interpolate between the extremes, the Euclidean ball being a kind of half-way point. To learn more about this spectrum of $m$s, with $m=2$ as a kind of midpoint between 1 and $\infty$, you could read about $L_p$ spaces in books on measure theory and functional analysis, and Banach space theory. In particular you could learn about how every $m$, which I am calling $p$, between 1 and 2 has a "twin" called $q$ which is between 2 and $\infty$, and $p$ and $q$ are so-called Holder conjugates, related by $\frac{1}{p} + \frac{1}{q} = 1$. To understand the relationship between the unit balls and the distances between points, center a copy of the unit ball at $X$, and shrink or expand it until it touches $Y$. See also "Minkowski functional" or "Gauge".

Another instructive exercise would be to plot the "unit sphere" for values of $m$ between 0 and 1, for instance, $m=\frac{1}{2}$. What do you notice is different about such spheres ? Why does this make them unsuitable for measuring distance ? Hint: Consider the norm of the sum of two vectors.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.