Solve the above program

Question

Consider the problem of covering the triangle with vertices at the points $(0, 0), (0, 1),$ and $(1, 0)$ with a ball of smallest radius. $$\min r$$ $$s. t. \> x ^2 + y ^2 ≤ r$$ $$(x − 1)^ 2 + y ^2 ≤ r$$ $$x^ 2 + (y − 1)^ 2 ≤ r.$$ Solve the above program

Answer 1

There’s no need for fancy techniques. Clearly the diameter of the disk must be at least the distance between the points $\langle 0,1\rangle$ and $\langle 1,0\rangle$, which is $\sqrt2$, so let’ see what happens if we try a disk of radius $\frac{\sqrt2}2$ centred midway between those points, at $\left\langle\frac12,\frac12\right\rangle$; does it cover $\langle 0,0\rangle$?