Lemma 3.7. Assuming that:

$φ : V \to ℝ$
$⟨ φ, 1 ⟩ = 0$

Then there is

ψ : V \to ℝ_{\geq 0}

such that

q_{{\tilde{L}}_{G}} (ψ) \leq q_{{\tilde{L}}_{G}} (ψ)

,

| supp ψ | = | {x : ψ (x) > 0} | \leq \frac{| V |}{Q}

and any threshold cut for

ψ

is a threshold cut for

φ

.