I can't read substack on my phone, so I can't see the article, but the correct statement that is closest to what you have written is just that d is any real number satisfying this inequality. We define a subset U of AxBxR by

U={(a,b,x):x>|a-b|^2}

and then were looking for the infimum of (the image of) U under the third coordinate function

d(a,b,x)=x