Coarea Inequality

If $n<m$ and $f:\mathbb{R}^n \to \mathbb{R}^m$ is an injective (and reasonably nice) function then image of $f$ is an $n$-dimensional object sitting inside the larger $\mathbb{R}^m$. Examples: a curve or surface in $\mathbb{R}^3$. But what if we turn the tables and consider functions $f:\mathbb{R}^m \to \mathbb{R}^n$ where $m > n$? There is just not enough room and many points must map to a common target point. In this talk I will explain the precise mathematical meaning of the following and end with the Coarea Inequality:

If $m>n$, and $f:\mathbb{R}^m \to \mathbb{R}^n$ is Lipschitz, then for almost every $y \in \mathbb{R}^n$, the set of points that map to $\{y\}$ is $(m-n)$-dimensional.