I guess you can create a function such as this:

minimax[table_] := Module[{slices,val,pos={}},
  slices = table[[All, All, #]] & /@ Range[Last@Dimensions@table];
  val = Min@Max@slices[[#]] & /@ Range[Last@Dimensions@table];
  pos = AppendTo[pos,val[#],2]

  ]

Then just run the

minimax[t] where t where t is your table.

EDIT: This way you will get the positions in the second level of your table. You could have also written

      pos= AppendTo[pos,Position[Min@Max@slices[[#]],slices[[#]]]] & /@ Range[Last@Dimensions@table];

I guess you can create a function such as this:

minimax[table_] := 
 Module[{slices, val, pos = {}}, 
  slices = table[[All, All, #]] & /@ Range[Last@Dimensions@table];
  val = Min@Max@slices[[#]] & /@ Range[Last@Dimensions@table];
  pos = Position[slices[[#]], val[[#]]] & /@ 
    Range[Last@Dimensions@table]
  ]

Then just run the

minimax[t] where t where t is your table.

EDIT: This way you will get the positions in the second level of your table.

EDIT2: cleared up the code this will give you pairs of indices, of the first two levels

I guess you can create a function such as this:

minimax[table_] := Module[{slices},
  slices = table[[All, All, #]] & /@ Range[Last@Dimensions@table];
  Min@Max@slices[[#]] & /@ Range[Last@Dimensions@table]
  ]

Then just run the

minimax[t] where t where t is your table.

I guess you can create a function such as this:

minimax[table_] := Module[{slices,val,pos={}},
  slices = table[[All, All, #]] & /@ Range[Last@Dimensions@table];
  val = Min@Max@slices[[#]] & /@ Range[Last@Dimensions@table];
  pos = AppendTo[pos,val[#],2]

  ]

Then just run the

minimax[t] where t where t is your table.

EDIT: This way you will get the positions in the second level of your table. You could have also written

      pos= AppendTo[pos,Position[Min@Max@slices[[#]],slices[[#]]]] & /@ Range[Last@Dimensions@table];