A minimum of three maximums Let $a,b,c,d$ be constants in the interval $[-1,1]$. Let $A$ be the minimum of the following three numbers:
$$\max\{2-a-c, |b-d|\},$$
$$\max\{2-b-c, 2-a-d, 2-b-d\},$$
$$\max\{2+b-c, 2-a+d, 2+b+d\}.$$
Is there a neater way to express $A$?

If each of $b,d$ are replaced by their negatives $-b,-d$ then the first max is unchanged, and the second and third max get switched, since the numbers considered get switched (in the same order as written). So you can include $b>d$ as an assumption, and thus drop the absolute value in the first max, making it $\max \{2-a-c,\ b-d \}$. (This isn't much of a simplification.)
