[?] Get the Maximum of a Minimum with Abs?

Questions
1

I am very beginner in Mathematica. I am trying to calculate the following

Maximize[{Minimize[{Abs[u - v], 3 <= u <= 5}, u], 7/2 <= v <= 6}, v]

but Mathematica does not give me any useful result.
Would you please tell me what is wrong?

2

It seems that Minimize does not work well with parameters and with Abs. Here is a workaroud:

Maximize[{MinValue[{Sqrt[(u - v)^2], 3 <= u <= 5}, u], 7/2 <= v <= 6},
  v]

The following result seems wrong:

MinValue[{RealAbs[u - v], 3 <= u <= 5}, u, Reals] // PiecewiseExpand

Piecewise[{{3 - v, u - v >= 0}}, -5 + v]
3

Thank you:)