Is it true that $$\inf_{t,x,y\in\mathbb{R}}\max\left(\left|x-1\right|+\left|tx-1\right|,\left|y-1\right|+\left|ty-2\right|\right)=1/3?$$
Thanks.
Is it true that $$\inf_{t,x,y\in\mathbb{R}}\max\left(\left|x-1\right|+\left|tx-1\right|,\left|y-1\right|+\left|ty-2\right|\right)=1/3?$$
Thanks.
Brute force should suffice here. Write $F(x,y,t) = \max\left(\left|x-1\right|+\left|tx-1\right|,\left|y-1\right|+\left|ty-2\right|\right)$. A moment's thought shows $F \ge 1/3$ outside the cube $[2/3, 4/3] \times [2/3,4/3] \times [1/2,2]$. You can divide this cube into simplices according to the signs of $x-1$, $tx-1$, $y-1$, $ty-2$; then in each simplex you are optimizing linear functions with respect to linear constraints, which is easy if tedious (assign it as a homework in a multivariable calculus course if necessary).