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 $n$; viewed as points in $\mathbb{R}^n$, 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.