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Level set estimates for strictly convex and hyperbolic functions
Abstract
In this note, we prove uniform upper bounds for the volume of the level set {x∈Ω: c≤ f(x)<c+δ}, c∈ℝ, δ>0, for strictly convex and hyperbolic functions f defined on a convex domain Ω⊂ℝn (n≥ 2). In particular, under a Hessian lower bound D2f≥ In, we obtain the sharp volume bound |Sn−1| r(Ω)n−2 δ, where r(Ω)=1/2 diamΩ. As an application, we derive L2 estimates for oscillatory integrals with convex phases.
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BibTeX
@misc{chen2026level,
title = {{Level set estimates for strictly convex and hyperbolic functions}},
author = {Xianghong Chen and Tongou Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32697},
eprint = {2609.32697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32697},
}
RIS
TY - UNPB
TI - Level set estimates for strictly convex and hyperbolic functions
AU - Xianghong Chen
AU - Tongou Yang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32697
UR - https://arxiv.org/abs/2609.32697
AB - In this note, we prove uniform upper bounds for the volume of the level set {x∈Ω: c≤ f(x)<c+δ}, c∈ℝ, δ>0, for strictly convex and hyperbolic functions f defined on a convex domain Ω⊂ℝ^n (n≥ 2). In particular, under a Hessian lower bound D^2f≥ I_n, we obtain the sharp volume bound |S^n−1| r(Ω)^n−2 δ, where r(Ω)=1/2 diamΩ. As an application, we derive L^2 estimates for oscillatory integrals with convex phases.
ER -