On bounded open convex domains, the first Dirichlet eigenfunction of the Laplacian and Ornstein-Uhlenbeck operator is shown to be α-logconcave for α ≤ 1/2 with explicit scaling thresholds, plus local convexity results and counterexamples for other operators.
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Establishes a D^{-3} lower bound on the fundamental gap for large horoconvex domains in hyperbolic space, matching a prior upper bound.
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To $1/2$-logconcavity and beyond: Geometric properties of Dirichlet eigenfunctions
On bounded open convex domains, the first Dirichlet eigenfunction of the Laplacian and Ornstein-Uhlenbeck operator is shown to be α-logconcave for α ≤ 1/2 with explicit scaling thresholds, plus local convexity results and counterexamples for other operators.