What are some practical usages of computing volume in $n$ dimensions?

Question

I am not sure if this is really a mathematical problem, but I know how to find volume of say a sphere in n dimensions, but after coming to realize how to do this, i just don't get what would be the practical usages of the result?

Like for instance, why would it be of any interest to find volumes of objects in multi dimensions, are there any practical reasoning or applications to this instead of looking from a purely mathematical point of view?

I know for a fact many mathematical theorems dont have any practical usage but could be just a mathematically interesting idea and possibly lead to new mathematical techniques in the future, but i was just curious what applications n dimensional volumes might have.

Answer 1

It seems to me that you might have the misconception that high-dimensional calculus or linear algebra may have dubious practical value because the world around us looks 3 (or 4) dimensional. The crucial point being misunderstood by this impression is that the main reason high-dimensional math arises in practical problems is because the problems involve many parameters. Each new parameter is often a new coordinate, or is its own dimension (degree of freedom).

For example, in computational chemistry the description of a molecule with many atoms is described by many coordinates at once: 3 each for the position of each atom, extra coordinates to describe the lengths of angles of chemical bonds, etc.

Another example of a practical problem where the framework for thinking about it uses very high dimensions is developing a recommendation system (like Netflix suggesting movies you might like). The description makes each user a separate dimension: that's a space with millions of dimensions! I'm not saying volume calculations are made here, but it's important to appreciate why high-dimensional settings show up first.

You want to know practical reasons for caring about high-dimensional volumes. If you need to integrate in high dimensions then you need a language to discuss volumes in high dimensions. High dimensional integration occurs in many areas, such as mathematical finance (see here), statistics (see here), and statistical mechanics (see here).

Answer 2

The real question here seems to me to be "What is the use of $n$ dimensions?" The history of mathematics from the $19$th century onwards has tremendous applications for $n$ dimensional space in physics. See, for example, Wikipedia article Phase space. One important idea is that given any list of $n$ numbers, it can be interpreted as the coordinates of a point in an $n$ dimensional space. This allows the use of geometric concepts such a volume and hyperplanes. This is the conceptual basis for Data science and Machine learning.