When Higher Dimensions Help

Question

I'm interested in examples of situations that are easier in higher dimensions. To give a flavor of what I am looking for, here are two of my favorites:

(a) In dimensions 2 and higher, one can characterize the standard normal distribution (up to a constant multiple) as the spherically symmetric distribution with independent marginals. Obviously, such a characterization fails miserably in one dimension.

(b) In dimension 3 and higher, you can prove Desargues' Theorem using the incidence axioms. No such proof works in two dimensions (and there are non-Desarguesian planes).

What are some other nice results where having at least $n$ dimensions allows one to prove things or characterize things in ways that are not possible in fewer than $n$ dimensions?

Answer 1

Here is an example which is easier for higher genus. The Four Color Theorem for the complex plane is much harder than the analogue result for surfaces with higher genus $g\ge 1$. In fact, the maximum number $p$ of colors needed depends on the (closed) surface's Euler characteristic $χ$ according to the formula $$ p=\left\lfloor {\frac {7+{\sqrt {49-24\chi }}}{2}}\right\rfloor . $$

Answer 2

The Poincaré conjecture is much easier to prove for its generalization to higher dimensions. Actually, the first proof was for dimension $n\ge 5$ by Smale in $1960$. Michael Freedman solved the case $n = 4$ in 1982 and received a Fields Medal in 1986. Grigori Perelman solved case $n = 3$ in 2003. This was still possible to prove, but barely so.