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I was just finished working on a project where we had to allocate facilities in a $2$-dimensional plane in order to satisfy certain demand restrictions while keeping the cost at minimum. I do NOT have a degree in mathematics so my apologies if my question may seem a little dumb and uneducated. We worked with cartesian coordinates and a distance function $$d:\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$$ with the distance between $(x_1,y_1)$ and $(x_2,y_2)$ given by $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. In the assignment it was specifically stated that we used the euclidean distance that I started asking myself: what other types of distances measurements could we use, I mean, like absolute deviation? This brings my to my question:

What other primary distance measure(s) exist(s) and what are possible applications of these?

I am curious to seewhat you guys can bring me!

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    I'm on my phone so I can't give too many details. Some interesting distances that I work with frequently include the Manhattan distance, Mahalanobis distance, and Hausdorf distance. Additionally, the KL divergence is interesting, but not a true metric. I will write more detail when I get home, but looking into those could give you a nice start. – 2017-01-12
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    great! can't wait for your extended answer. – 2017-01-12

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There is a set of related possible distance measures (which some call "metrics") of the form

$$d(a,b) = \left( |(a_x-b_x)|^\ell + (a_y-b_y)|^\ell \right)^\frac1{\ell}$$

The distance between a point and the origin in such a metric is often called the "$\ell$-norm, and when $\ell = 2$ you get the Euclidean metric, while as $\ell$ goes to infinity, you get a metric which is the larger of the $x$ or $y$ separations.

WHen $\ell = 1$ you get the sum of $|(a_x-b_x)| + |(a_y-b_y)|$, sometimes called the "Manhatan metric" because it is the distance a taxi would have to travel on orthogonal streets.

But there are plenty of other metrics as well. One property that they all share is the triangle identity:

$$d(a,b) + d(b,c) \geq d(a,c)$$

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    You mean the triangle inequality, right? – 2017-01-12
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The term you're looking for is a metric. From the wiki: A metric or distance function is a function that defines a distance between each pair of elements of a set.

A fun example is the so-called "taxicab metric", or $L^1$, where the distance function (or metric) is given by

$$d(\mathbf{x,y})=\sum_{i=1}^n|x_i-y_i|,$$ where $\bf{x,y}$ are $n$-dimensional vectors. $L^2$ is the usual Euclidean metric (here is a link on $L^p$) and there are many others. PBS Infinite Series recently made a nice video on the subject.