In a seminal paper, Kannan and Lovász (1988) considered a quantity $μ_{KL}(Λ,K)$ which denotes the best volume-based lower bound on the covering radius $μ(Λ,K)$ of a convex body $K$ with respect to a lattice $Λ$. Kannan and Lovász proved that $μ(Λ,K) \leq n \cdot μ_{KL}(Λ,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log(2n))$ factor suffices, which would match the lower bound from the work of Kannan and Lovász. We settle this conjecture up to a constant in the exponent by proving that $μ(Λ,K) \leq O(\log^{3}(2n)) \cdot μ_{KL} (Λ,K)$. Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a $(\log(2n))^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal flatness constant of $O(n \log^{2}(2n))$, improving on the previous bound of $O(n^{4/3} \log^{O(1)} (2n))$.

翻译：在一篇开创性论文中，Kannan 和 Lovász (1988) 考虑了量 $μ_{KL}(Λ,K)$，它表示相对于格 $Λ$ 的凸体 $K$ 的覆盖半径 $μ(Λ,K)$ 的最佳基于体积的下界。Kannan 和 Lovász 证明了 $μ(Λ,K) \leq n \cdot μ_{KL}(Λ,K)$，而 Dadush (2012) 提出的子空间平坦性猜想认为 $O(\log(2n))$ 因子就足够了，这将匹配 Kannan 和 Lovász 工作中的下界。我们通过证明 $μ(Λ,K) \leq O(\log^{3}(2n)) \cdot μ_{KL} (Λ,K)$，在指数范围内解决了该猜想直至一个常数因子。我们的证明基于 Regev 和 Stephens-Davidowitz (2017) 的逆向闵可夫斯基定理。沿用 Dadush (2012, 2019) 的工作，我们获得了求解 $n$ 变量整数规划的 $(\log(2n))^{O(n)}$ 时间随机算法。我们主要结果的另一个推论是近乎最优的平坦性常数 $O(n \log^{2}(2n))$，改进了先前 $O(n^{4/3} \log^{O(1)} (2n))$ 的界。