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arXiv · 2609.32697

Level set estimates for strictly convex and hyperbolic functions

Xianghong Chen·Tongou Yang
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Abstract

In this note, we prove uniform upper bounds for the volume of the level set $$\{x\inΩ: c\le f(x) 0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $Ω\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n-1}|\, r(Ω)^{n-2}\,δ,$$ where $r(Ω)=\frac12\,{\mathrm{diam}Ω}$. As an application, we derive $L^2$ estimates for oscillatory integrals with convex phases.

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Xianghong Chen, Tongou Yang. 2026-09-26. Level set estimates for strictly convex and hyperbolic functions. https://arxiv.org/abs/2609.32697

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