I was reading this PDF that proves the Hopf-Lax Formula, but I can't figure out why the sentence at the end of page $5$ is true. It says
Thus$$tL\left (\frac{\mathbf{x}-\mathbf{y}}{t}\right )+g(\mathbf{y})\geq u(\mathbf{x},t)$$if $|\mathbf{x}-\mathbf{y}|\geq tB$, for$$B\equiv \max \left [A,\frac{L(0)}{\operatorname{Lip}(g)+1}\right ].$$Hence $(12)$ becomes$$u(\mathbf{x},t)=\min \limits _{y\in B_{tB}(\mathbf{x})}\left \{tL\left (\frac{\mathbf{x}-\mathbf{y}}{t}\right )+g(\mathbf{y})\right \}.$$
I understand everything before that point, but I'm not sure why if $tL\left (\frac{\mathbf{x}-\mathbf{y}}{t}\right )+g(\mathbf{y})\geq u(\mathbf{x},t)$ for $\mathbf{y}\not \in B_{tB}(\mathbf{x})$ then the infimum becomes an infimum over the ball (it is compact so the infimum then becomes a minimum). At first I thought that it should be something like
If $A\subset \mathbb{R}^n$ and $f(x)>\inf \limits _{y\in A}f(y)$ for all $x\not \in A$ then $\inf \limits _{x\in \mathbb{R}^n}f(x)=\inf \limits _{x\in A}f(x)$.
But this isn't true, so I have no clue about how to prove that last assertion. Can someone help me with this? Thanks in advance.
