ScalingStacks

Lemma 8.6 . [02Y3]

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Lemma 8.6.

Let ψ:ℝ→ℝ\psi\colon\mathbb{R}\to\mathbb{R} be a concave function whose stability set is an interval [a,b][a,b]. Then

2​∫ℝψ∨∘∂ψ​ℳℤ​(ψ)=(b−a)​(ψ∨​(a)+ψ∨​(b))+∫ℝ(ψ′​(u)−a)​(b−ψ′​(u))​d​u.2\int_{\mathbb{R}}\psi^{\vee}\circ\partial\psi\,{\mathcal{M}}_{\mathbb{Z}}(\psi)=(b-a)(\psi^{\vee}(a)+\psi^{\vee}(b))+\int_{\mathbb{R}}(\psi^{\prime}(u)-a)(b-\psi^{\prime}(u))\,\text{\rm d}u.

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