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I have trying to learn The isoperimetric problem (http://mathworld.wolfram.com/IsoperimetricProblem.html) However it seems the idea of volume and surface area is to extend to Gaussian spaces as described in https://arxiv.org/pdf/1202.4124.pdf

I can understand the volume, surface area of an object in 3 dimensional, but how it is extended to a set? What background is required to understand this extension? Any suggestion shall be appreciated.

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At the high-end of mathematics, this kind of problem is studied in the branch of "sub-Riemannian Geometry" which is kind-of-like Riemannian Geometry, but weirdly different.

However, your question seems much more basic: you first need to understand the concept of spaces in more than 3 dimensions, and then to understand the concept of of (smooth) manifolds, and then some basic concepts relating boundaries to the interiors they enclose. This leads naturally to "calculus on manifolds", which gives explicit formulas that relate volume, area, perimeter in various general settings. Its also possible to do all this in 3 or fewer dimensions, but the general n-dimensional case really is not any harder; even easier, in a way.

Almost all "sets" do not even have a boundary -- a general set might be like a sprinkling of sand, maybe with some sharp pointy bits going off to infinity, other parts shaped like nasty fractals or worse - sets in general aren't even "measurable".

I suspect you'll get bored trying to read the standard college-level pre-requisites for this ("multivariate calculus" and "linear algebra"), so you might instead enjoy picking through nearby topics, such as "set theory", "general topology", "measure theory". The danger is you'll get stuck on advanced topics with a very weak basic grounding, though. The basic grounding is super-important to have, although learning it can be boring (depending on the text).

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  • $\begingroup$ I was expecting a basic definition of volume and surface area for applicable sets in English if possible (rather than in mathematical notation) $\endgroup$
    – Creator
    Feb 10, 2018 at 18:29
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    $\begingroup$ The "applicable sets" here are "Borel sets", and these are, by definition exactly those sets that can have have an area (in 2D) or a volume (in 3D). The area/volume of a Borel set is called its "measure" and the Banach Tarski paradox infamously shows what happens if you work with sets that are not measurable. The area/volume of a Borel set is more-or-less exactly what you might intuitively guess it might be, if you were a high-school student. "Measure theory" is the relevant branch. Try "Probability Theory - A Comprehensive Course | Achim Klenke | Springer", chapters 1,2. $\endgroup$
    – Linas
    Feb 11, 2018 at 8:46
  • $\begingroup$ A "measure" is a certain way of consistently assigning a size to any set that isn't too crazy. Given a collection of Borel sets, there's an infinite number of ways to assign a measure to them. The Euclidean measure is the high-school intuitive one, for sets in a plane or 3D space. The Gaussian measure is just the bell curve, which just says "the rules of the game are that a set gets smaller, the farther it is from the origin". $\endgroup$
    – Linas
    Feb 11, 2018 at 8:57

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