Calculate the minimum absolute difference between the maximum and minimum number from the triplet of 3 arrays

Question

Question is:

Calculate the minimum absolute difference between the maximum and minimum number from the triplet a, b, c such that a, b, c belongs arrays A, B, C respectively.

There is another question: Find i, j, k such that : max(abs(A[i] - B[j]), abs(B[j] - C[k]), abs(C[k] - A[i])) is minimized. Return the minimum max(abs(A[i] - B[j]), abs(B[j] - C[k]), abs(C[k] - A[i])). A, B & C are sorted arrays.

Reading the solution approach here and here, I realize that the solution approach for both these questions is same, to:

minimize abs( max(a,b,c) - min(a,b,c) )

I'm not sure why this follows and how we should go about relating these two questions and their algorithms. Why are these two questions identical?

Answer 1

Can you see the following equality?
abs( max(a,b,c) - min(a,b,c) ) = max(abs(a - b), abs(b - c), abs(c - a))

Here is a hint to prove the equality.

Without loss of generality, we can assume a >= b >= c.

By the way, the first abs is not necessary since max(a, b, c) is always no less than min (a, b, c).