ScalingStacks

Proposition A.1 . [01HN]

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Proposition A.1.

There exists C>0C>0 such that every Lipschitz continuous convex function φ:τ→𝐑\varphi:\tau\to\mathbf{R} satisfies

C−1​‖φ‖C0,1​(τ)≤‖φ‖C0​(∂τ)+supe∈ℰ⁡(τ),v∈int⁡(τ)|Dπe​(v)​φ​(e)|≤C​‖φ‖C0,1​(τ).C^{-1}\|\varphi\|_{C^{0,1}(\tau)}\leq\|\varphi\|_{C^{0}(\partial\tau)}+\sup_{e\in\mathcal{E}(\tau),v\in\mathrm{int}(\tau)}\left|D_{\pi_{e}(v)}\varphi(e)\right|\leq C\,\|\varphi\|_{C^{0,1}(\tau)}.

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