Let $u$ be a continuous function in an open set $\Omega\subset \mathbb{R}^{n}$, and let $H$ an open set such that $\bar{H}\subset \Omega$. We define , for $\epsilon >0$, the upper $\epsilon$-envelope of $u$ (with respect to $H$):
$u^{\epsilon}(x_{0})=\sup_{x \in \bar{H}}\Big\{u(x)+\epsilon -\frac{1}{\epsilon}|x-x_{0}|^{2}\Big\}$
For $x_{0} \in H$. The autor says: the graph of $u^{\epsilon}$ is the envelope of the graphs family $\{P^{\epsilon}_{x}\}_{x \in \bar{H}}$ of concave paraboloids of opening $2/\epsilon$ and vertex $(x,u(x)+\epsilon)$.
First, whats is the envelope of the family of graphs? I considered the one dimensional case where $u(x)=x$, $x\in \mathbb{R}$, the concave "paraboloid when $x_{0}=1$, for example is $P^{\epsilon}(x)=-\epsilon.(x-1)^{2}+\epsilon +x$. I used a program and draw the graph, I did vary the value of Epsilon, and I saw that the paraboloid is gradually crossing the whole line. But, but i still do not understand the geometry of that definition