The distance between point and line independently on metric is defined by $$d(X, l) = \inf\{d(X, Y)|Y\in l\}.$$ I have troubles understandning how this infimum works.
Can someone please give me an explanation why it is defined like this and how it works with examples and explanations in $\mathbb{R}^2$ for euclidean distance $d_e((x_1, x_2), (y_1, y_2)) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$, maximum distance $\max\{|x_1 - y_1|, |x_2 - y_2|\}$ and manhattan distance $|x_1 - y_1| + |x_2 - y_2|$?
I'm having really hard time understanding this.
Many thanks!