Let $S = [0, 1] × [0, 1]$. Which of the following statements are true?

Question

If $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ are arbitrary points in the plane, define the metric $d(P, Q) = max\{|x_1 − x_2|, |y_1 − y_2|\}.$ Let $P = (2,\frac{1}{2})$. Let $S = [0, 1] × [0, 1]$.

Which of the following statements are true?

$a.$ There does not exist any point $Q ∈ S$ such that $d(P, Q) = \min\{d(P, X) | X ∈ S\}.$

$b.$ There exists a unique point $Q ∈ S$ such that $d(P, Q) = \min\{d(P, X) | X ∈ S\}.$

$c.$ There exist infinitely many points $Q ∈ S$ such that $d(P, Q) = \min\{d(P, X) | X ∈ S\}$

I thinks option $b)$ is correct

Any hints/solution

thanks u

Answer 1

a. is false: $Q = \left(1,\frac{1}{2}\right)$ is such a point (the minimum being 1).
b. is false: $Q = \left(1,1\right)$ is such a point (and I have no idea what $f$ is supposed to have to do with anything.
c. is true: $Q = \left(1,x\right)$ is such a point for every $x \in [\frac{-1}{2},1]$.