In a seminal paper, Kannan and Lov\'asz (1988) considered a quantity $\mu_{KL}(\Lambda,K)$ which denotes the best volume-based lower bound on the covering radius $\mu(\Lambda,K)$ of a convex body $K$ with respect to a lattice $\Lambda$. Kannan and Lov\'asz proved that $\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)$ 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 Lov\'asz. We settle this conjecture up to a constant in the exponent by proving that $\mu(\Lambda,K) \leq O(\log^{3}(2n)) \cdot \mu_{KL} (\Lambda,K)$. 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 $O(n \log^{3}(2n))$.

翻译：在一篇开创性论文中，Kannan 与 Lovász（1988）考虑了量 $\mu_{KL}(\Lambda,K)$，该量表示凸体 $K$ 关于格 $\Lambda$ 的覆盖半径 $\mu(\Lambda,K)$ 的最佳基于体积的下界。Kannan 和 Lovász 证明 $\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)$，而 Dadush（2012）提出的子空间平坦性猜想声称 $O(\log(2n))$ 因子即足够，这将匹配 Kannan 和 Lovász 工作的下界。我们通过证明 $\mu(\Lambda,K) \leq O(\log^{3}(2n)) \cdot \mu_{KL} (\Lambda,K)$ 将这一猜想解决至指数内的常数因子。我们的证明基于 Regev 与 Stephens-Davidowitz（2017）提出的逆向闵可夫斯基定理。遵循 Dadush（2012, 2019）的工作，我们得到一个 $(\log(2n))^{O(n)}$ 时间的随机算法，用于求解 $n$ 变量整数规划。我们主要结果的另一个推论是接近最优的平坦常数 $O(n \log^{3}(2n))$。