Near-Optimal Oracle Bounds for Isotropic Rounding

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 $\Cov(\Unif(K))\preceqκI_n$, with $κ\ge1$, then approximate uniform sampling from an initial density bounded by a constant times the uniform density uses $\Ot(n^2\sqrtκ)$ expected membership queries. We prove this by simulating reflected kinetic dynamics using the analysis of Eberle and Lörler~\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.