Maybe this is a stretch, but laws of large numbers, and more precise concentration of measure bounds, can be proven by the "isoperimetric" approach which can be called **geometric** and was pioneered by Talagrand.

For example, consider $n$ independent fair coin flips, i.e. the set of binary sequences of length $n$; we can see that a law of large numbers is roughly equivalent to the statement that almost all of these sequences fall in a ball of small radius as $n \to \infty$.

More generally, Talagrand states that in a product measure space, if $A$ has large measure, then the measure of points within distance $\epsilon$ of $A$ (i.e. $A$ plus its perimeter) is *very* large.

References: E.g. ["Notes on Talagrand's Isoperimetric Inequality" by Nick Cook][1].


  [1]: https://services.math.duke.edu/~nickcook/talagrand.pdf