Examples of results failing in higher dimensions A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging.
One of the examples of rigor-lacking approach are proofs done via graphs or pictures without formalizing the reasoning. I would like thus to come up with a few examples of theorems (or other important results) which may be true in low dimensions (and are pretty intuitive graphically) but fail in higher dimensions.
By the way, these examples are directed towards people who do not have a strong mathematical background (some linear algebra and calculus), so avoiding technical statements would be appreciated.
Jordan-Schoenflies theorem could be such an example (though most economists are unfamiliar with the notion of a homeomorphism). Could you point me to any others?
Thanks.

Here's an example that doesn't require too much mathematical knowledge, and the low-dimensional result is intuitive graphically:
We know that if a differentiable function $ f : \mathbb{R} \to \mathbb{R} $ has only one stationary point, which is a local minimum, then it must be a global minimum (this is intuitively obvious, and can be proved using Rolle's theorem). However, this result does not generalise to higher dimensions. An example would be $f : \mathbb{R}^2 \to \mathbb{R} $ with $ f(x,y) = x^2 + y^2(1-x)^3 $. This function has a unique stationary point at $ (0,0) $, which is a local minimum but not a global minimum (this can be seen by considering $ x >> 1 $). 
(Interactive 3D plot)
