{"id":"hexagon/2610.00726","pdf":"https://hexagonmath.org/pdf/2610.00726v1","url":"https://hexagonmath.org/abs/2610.00726v1","tags":["math.MG","math.OC","msc:15A60","msc:15B48","msc:47A63","msc:46B70","msc:49Q22","Bures-Wasserstein distance","Lin-van den Driessche conjecture","Schatten norm"],"score":0,"title":"The $p$-Bures–Wasserstein distance via $3\\times3$ block matrices (with application to $p$-Bures–Wasserstein barycenters)","domain":"math.FA","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.FA","summary":"We study the $p$-Bures–Wasserstein distance $D_p(A,B)=(\\mathrm{tr}\\,A^p+\\mathrm{tr}\\,B^p-2\\|A^{1/2}B^{1/2}\\|_p^p)^{1/2}$ on positive semidefinite matrices; it equals the Bures–Wasserstein distance for $p=1$, and the Hilbert–Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\\le p\\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\\times3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}