We began with the postulate and constraints
postulate:=
p
∑
i=1
Abs[
x
i
-
y
i
]
p
∑
i=1
Max[
x
i
,
y
i
]
1-
p
∑
i=1
Min[
x
i
,
y
i
]
p
∑
i=1
x
i
+
p
∑
i=1
y
i
-
p
∑
i=1
Min[
x
i
,
y
i
]
;​​constraints:=(p∈)∧(0<p)∧(i∈)∧(1≤i≤p)∧(x∈Arrays[p,

>0
])∧(y∈Arrays[p,

>0
]);
John Hendrickson has given us an interesting relation and the hint to use induction.
So putting the sums on the lhs, we have
unknown:=
p
∑
i=1
Abs[
x
i
-
y
i
]
p
∑
i=1
Max[
x
i
,
y
i
]
+
p
∑
i=1
Min[
x
i
,
y
i
]
p
∑
i=1
x
i
+
p
∑
i=1
y
i
-
p
∑
i=1
Min[
x
i
,
y
i
]
;
For p  1 we have John’s relation
In[]:=
Abs[x1-y1]
Max[x1,y1]
+
Min[x1,y1]
x1+y1-Min[x1,y1]
//FullSimplify
Out[]=
1
For p  2:
In[]:=
Abs[x1-y1]+Abs[x2-y2]
Max[x1,y1]+Max[x2,y2]
+
Min[x1,y1]+Min[x2,y2]
x1+x2+y1+y2-(Min[x1,y1]+Min[x2,y2])
//FullSimplify
Out[]=
1
For p  3
Abs[x1-y1]+Abs[x2-y2]+Abs[x3-y3]
Max[x1,y1]+Max[x2,y2]+Max[x3,y3]
+
Min[x1,y1]+Min[x2,y2]+Min[x3,y3]
x1+x2+x3+y1+y2+y3-(Min[x1,y1]+Min[x2,y2]+Min[x3,y3])
//FullSimplify
... still running! But perhaps there’s hints from p  2 for an induction proof.