Distance in metric space p_{1}

Question

I need to evaluate distance of point [6,6] and circle $x^2 + y^2 = 25$ in metric space $p_{1}(x,y) = ∑|x_k-y_k|$ (sum metric). I know that I need to count $inf(p_{1}([6,6],X), X $ are points from circle, but I: 1) don't know how to write this as function; 2) can't derivate it to find minimum; 3) don't know if I am thinking in the right way.

Answer 1

Well, it's clear the closest point is in the positive quadrant. So we need to minimise $d(x,y) = (6-x) + (6-y)$ (the distance from $(6,6)$ to $(x,y)$ on the circle) under the constraint $x^2 + y^2 = 25$. Of course we can just maximise $x+y$ instead. Intuitively it's clear this should happen on the line $y = x$, which is confirmed by applying the Lagrange multiplier method, e.g. So the point $(\sqrt{\frac{25}{2}},\sqrt{\frac{25}{2}})$ assumes the minimum (which exists by compactness).