Abstract
Let \(\phi: \mathbb{R}^{n} \rightarrow \mathbb{R}\) be a convex function.
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Notes
- 1.
For the first inequality the doubling property is not used: if α ≥λ n , then ω n (|minΩ ϕ|∕2diam (Ω)) n ≤μ(αΩ) by Lemma 3.2.3. If α < λ n , then |minΩ ϕ| n < (1 − α) (2diam (Ω)) n c n μ(Ω) by definition of λ n.
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Gutiérrez, C.E. (2016). The Cross-Sections of Monge–Ampère. In: The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications, vol 89. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43374-5_3
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