An affine reverse Faber-Krahn inequality via information decay
Abstract
We prove that every convex body K⊂ℝn admits S∈SLn(ℝ) such that λ1(SK)|K|2/n≤ Cn, where C is universal. For origin-symmetric bodies of volume one, the bound holds in maximal Gaussian measure position. The main step is the spectral estimate λ1(D)≤ C(P)(n+√(n I(ν|γn))) for an even isotropic log-concave probability measure ν supported on a symmetric convex body D, with Poincaré constant at most P and finite relative Fisher information. A second-order Stein estimate and an information jump under convolution give entropy decay faster than m−1 for normalised sums. Their supports control the Gaussian measure of dilates, and a Brownian bridge expansion gives the spectral estimate. A Gaussian-weighted ground state supplies a measure whose Fisher information is bounded in terms of λ1(K). The square-root dependence then gives the bound of order n.
Related
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BibTeX
@misc{colbrook2026affine,
title = {{An affine reverse Faber-Krahn inequality via information decay}},
author = {Matthew J. Colbrook and George Stepaniants},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02279},
eprint = {2611.02279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02279},
}
RIS
TY - UNPB
TI - An affine reverse Faber-Krahn inequality via information decay
AU - Matthew J. Colbrook
AU - George Stepaniants
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02279
UR - https://arxiv.org/abs/2611.02279
AB - We prove that every convex body K⊂ℝ^n admits S∈SL_n(ℝ) such that λ_1(SK)|K|^2/n≤ Cn, where C is universal. For origin-symmetric bodies of volume one, the bound holds in maximal Gaussian measure position. The main step is the spectral estimate λ_1(D)≤ C(P)(n+√(n I(ν|γ_n))) for an even isotropic log-concave probability measure ν supported on a symmetric convex body D, with Poincaré constant at most P and finite relative Fisher information. A second-order Stein estimate and an information jump under convolution give entropy decay faster than m^−1 for normalised sums. Their supports control the Gaussian measure of dilates, and a Brownian bridge expansion gives the spectral estimate. A Gaussian-weighted ground state supplies a measure whose Fisher information is bounded in terms of λ_1(K). The square-root dependence then gives the bound of order n.
ER -