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Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series

A mix of human-written and AI-generated textHuman understanding: all partsmath.PR — Probabilitymath.FA — Functional Analysismath.MG — Metric Geometry

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let X=∑i=1dgiBiX=\sum_{i=1}^d g_iB_i, where g∼N(0,Id)g\sim N(0,I_d), the matrices Bi∈Rd×dB_i\in\mathbb{R}^{d\times d} are symmetric, and ∑iBi2⪯Id\sum_iB_i^2\preceq I_d. For every even log-concave f ⁣:R→[0,∞)f\colon\mathbb{R}\to[0,\infty), we obtain Edet⁡f(X)≥(Ef(κG))d\mathbb{E}\det f(X)\ge(\mathbb{E} f(\kappa G))^d at dilation κ=13/5\kappa=13/5, where GG is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below π/2\sqrt{\pi/2} can work; we conjecture that this value is optimal.

Provenance statement

The results in this paper were developed through extensive use of AI assistants: OpenAI's GPT Pro for the main line of argument, including Theorem 2.1, the verification of its scalar inequalities, and Theorem 4.3; and Anthropic's Claude for the results under an ellipsoidal norm bound in Appendix E.3, Proposition 3.3, Appendix D.3, and the writing, positioning, and literature search. None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs.

Tools used

OpenAI
GPT Pro
Anthropic
Claude

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026