15.
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
y = (y1, ⋯, yn)
yi = xi +
ϵ
|S|
, if i ∈ S
= v(i) +
α
(n − |S|)
, if i ∈ NS
∑
i∈N
yi =
∑
i∈S
yi +
∑
j∈NS
yj =
∑
i∈S
(xi +
ϵ
|S|
) +
∑
j∈NS
(v(j) +
α
(n − |S|)
)
=
∑
i∈S
xi +
∑
i∈S
ϵ
|S|
+
∑
j∈NS
v(j) +
∑
j∈NS
α
(n − |S|)
=
∑
i∈S
xi +
ϵ
|S|
|S| +
∑
j∈NS
v(j) +
α
(n − |S|)
(n − |S|) =
∑
i∈S
xi + ϵ +
∑
j∈NS
v(j) + α =
∑
i∈S
xi + v(S) −
∑
i∈S
xi +
∑
j∈NS
v(j) + α
= v(S) +
∑
j∈NS
v(j) + v(N) − v(S) −
∑
j∈NS
v(j) = v(N)
16.
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
y = (y1, ⋯, yn)
yi = xi +
ϵ
|S|
, if i ∈ S
= v(i) +
α
(n − |S|)
, if i ∈ NS
i ∈ S yi = xi +
ϵ
|S|
≥ v(i) +
ϵ
|S|
> v(i)
i ∈ NS yi = v(i) +
α
(n − |S|)
≥ v(i)
α ≥ 0
ϵ > 0
17.
(N, ν)
∑
i∈S
x(i) ≥ v(S), ∀S ⊂ N ( * )
x S ( * )
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
α ≥ 0
y = (y1, ⋯, yn)
yi = xi +
ϵ
|S|
, if i ∈ S
= v(i) +
α
(n − |S|)
, if i ∈ NS
i ∈ S yi > xi
∑
i∈S
yi =
∑
i∈S
(xi +
ϵ
|S|
) =
∑
i∈S
xi +
∑
i∈S
ϵ
|S|
=
∑
i∈S
xi + ϵ =
∑
i∈S
xi + v(S) −
∑
i∈S
xi = v(S)
y domS x x