{"id":"hexagon/2610.00718","pdf":"https://hexagonmath.org/pdf/2610.00718v1","url":"https://hexagonmath.org/abs/2610.00718v1","tags":["math.FA","math.MG","msc:60B20","msc:60E15","msc:15B52","msc:52A40","determinant inequality","log-concave functions","matrix Gaussian series","random matrices"],"score":0,"title":"Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.PR","summary":"We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let $X=\\sum_{i=1}^d g_iB_i$, where $g\\sim N(0,I_d)$, the matrices $B_i\\in\\mathbb{R}^{d\\times d}$ are symmetric, and $\\sum_iB_i^2\\preceq I_d$. For every even log-concave $f\\colon\\mathbb{R}\\to[0,\\infty)$, we obtain $\\mathbb{E}\\det f(X)\\ge(\\mathbb{E} f(\\kappa G))^d$ at dilation $\\kappa=13/5$, where $G$ 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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro","Claude"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}