Near-Optimal Oracle Bounds for Isotropic Rounding

2026-10-05

AI Summary

This study investigates the query complexity of isotropic rounding for convex bodies. By integrating reflection dynamics simulation, approximate uniform sampling, and polyhedral geometric analysis, it achieves an efficient transformation of convex bodies into near-isotropic position. The core contributions are twofold: first, it establishes an O(n³) upper bound and an Ω(n³) lower bound, attaining optimal query complexity up to polylogarithmic factors; second, it constructs a matching lower bound proof that refutes prior conjectures concerning ellipsoidal growth. Collectively, these results provide a tight characterization of the computational complexity inherent in high-dimensional convex geometry.

Abstract

We give optimal bounds, up to polylogarithmic factors, for rounding convex bodies to near-isotropic position. A convex body $B(0,r)\subseteq K\subseteq B(0,R)$ in $\R^n$ can be rounded using $\Ot(n^3)$ membership queries; the upper bound extends to logconcave distributions. We prove a matching $\Omegat(n^3)$ lower bound, even when $R/r=n^{O(1)}$. The key lemma states that if $B(0,1)\subseteq K$ and $\Cov(\Unif(K))\preceq\kappa I_n$, with $\kappa\ge1$, then approximate uniform sampling from an initial density bounded by a constant times the uniform density uses $\Ot(n^2\sqrt\kappa)$ expected membership queries. We prove this by simulating reflected kinetic dynamics using the analysis of Eberle and L\"orler~\cite{EL26:journal}. Combining this sampler with the ideas of Jia, Laddha, Lee, and Vempala~\cite{JLLV26:journal} for rounding well-rounded bodies yields our $\Ot(n^3)$ query rounding algorithm. We also give a counterexample to an ellipsoid-growth conjecture from previous papers on this topic.

Problem

• Isotropic Rounding
• Convex Bodies
• Membership Queries
• Query Complexity
• Logconcave Distributions

Innovation

• Isotropic Rounding
• Membership Queries
• Reflected Kinetic Dynamics
• Logconcave Distributions
• Convex Bodies