How do I prove that the area of a sphere is the least possible area for a given volume?

Question

Or, why do soap bubbles have the shape of a sphere and not that of unicorns?

What's the math behind it?

Answer 1

First fact: the isoperimetric inequality in higher dimensions follows from the Brunn-Minkowski inequality.

Second fact: the shape of a bubble depends on the shape that minimize the potential, i.e. maximize the volume for a given area, or (dual problem) minimize the area for a given volume. By using reflections we have that any solution (we do not know in advance that there is a unique solution) is given by a convex shape, and unicorns are not convex bodies.

Answer 2

The maths behind this is a topic called "Calculus of Variations" - I would start there and you could do worse than something like Gelfand's book on the subject

You want to find the functional which will represent the surface of minimum area. This is a 2-D extension of simpler problems like finding minimal curves. For example, this is one of the ways you can derive the equation of a catenary - the shape that a heavy chain makes hanging from two points under its own weight.