An attaining coupling and potential on a three-point metric space

The specified distances are symmetric, vanish exactly on equal points, and are positive for distinct points. For the three distinct points, the triangle inequalities are $2\le 1+1$, $1\le 2+1$, and $1\le 1+2$; when a point repeats, each inequality is $0\le d+d$ or $d\le d+0$. Thus this is a metric. The displayed matrix is nonnegative, has row sums $(1\mathrm{/}2,0,1\mathrm{/}2)$ and column sums $(0,1,0)$, and has cost $(1\mathrm{/}2)\cdot 1+(1\mathrm{/}2)\cdot 1=1$.

The vector $f=(0,-1,0)$ has $f_1=0$. Its pairwise differences have magnitudes $1,1,0$, bounded respectively by distances $1,1,2$. Its objective is $(1\mathrm{/}2)\cdot 0-1\cdot (-1)+(1\mathrm{/}2)\cdot 0=1$.

The feasible plan and potential in steps 1.1 and 1.2 have the same value. By [F1], the primal optimum is at most the plan cost, the dual optimum is at least the potential value, and the two optima are equal; hence both are attained with value $1$. The positive entries are ${\pi }_{12}={\pi }_{32}=1\mathrm{/}2$, and they satisfy $f_1-f_2=d(x_1,x_2)=1$ and $f_3-f_2=d(x_3,x_2)=1$, as complementary slackness requires.