## Definition of $\epsilon$-upper envelope of a function

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

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