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 $O(\log n)$ 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 $\mu(\Lambda,K) \leq O(\log^{7}(n)) \cdot \mu_{KL} (\Lambda,K)$. 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 $(\log n)^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal flatness constant of $O(n \log^{8}(n))$.

翻译：在一篇开创性论文中，Kannan 和 Lovász（1988）考虑了一个量 \(\mu_{KL}(\Lambda,K)\)，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 证明了 \(\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)\)，而 Dadush（2012）提出的子空间平坦性猜想断言，一个 \(O(\log n)\) 的因子就足够了，这将匹配 Kannan 和 Lovász 工作中的下界。我们通过证明 \(\mu(\Lambda,K) \leq O(\log^{7}(n)) \cdot \mu_{KL} (\Lambda,K)\)，将这一猜想解决到指数上的常数因子内。我们的证明基于 Regev 和 Stephens-Davidowitz（2017）的逆向闵可夫斯基定理。沿着 Dadush（2012, 2019）的工作，我们得到了一个 \(n\) 变量整数规划的 \((\log n)^{O(n)}\)-时间随机算法。我们主要结果的另一个含义是得到近乎最优的平坦性常数 \(O(n \log^{8}(n))\)。