I'm reading a book and it states the following lemma:

Let $c \in \mathbb{R}, u: E \rightarrow [c, \infty]$ not identically equal to $\infty$ and define for $t > 0$: $$u_t(x) = \inf \{u(y) + t d(x,y) | y \in E\}$$ where $d$ is the distance in the metric space $X$. Then $\text{Lip}(u_t) \leq t, u_t \leq u$ and $u_t(x)$ increases to $u(x)$ as $t \rightarrow \infty$ whenever $x$ is a lower semicontinuous point of $u$.

I'm just having trouble with the notation:

  1. What is the $\inf$ being taken over? If it's all $y \in E$ doesn't second term become zero -- so that I would always find $u_t(x) = \inf \{u(y) + t d(x,y) | y \in E\} = u(x)$? I guess, generally, I'm having trouble understanding what this function is.

  2. What is $\text{Lip}(u_t)$? This was just some instantaneous notation.

  • $\begingroup$ okay #2 is just the lipschitz constant of $u_t$, how about #1? $\endgroup$ – yoshi Feb 13 at 17:45

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