A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. (English) Zbl 1128.35339
Summary: We prove a Brunn-Minkowski-type inequality for the eigenvalue \(\Lambda\) of the Monge-Ampère operator: \(\Lambda^{-1/2n}\) is concave in the class of \(C_+^2\) domains in \(\mathbb R^n\) endowed with Minkowski addition. The equality case is explicitly described too. The main device of the proof is a notion of addition for convex functions, called infimal convolution, which corresponds to the Minkowski addition of the graphs of the involved functions.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 26D15 | Inequalities for sums, series and integrals |
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