Suppose we have a function $f : \mathbb{R}^N \rightarrow \mathbb{R}$ which, given a vector, returns the value of its smallest element. How can I approximate $f$ with a differentiable function(s)? 



A differentiable approximation to the minimum function
If signs aren't a big deal, use the generalized mean formula


$$
\left(\frac{1}{n}\sum x\_i^k\right)^{1/k}
$$


for $k\to -\infty$.


