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 $\mu(\Lambda,K) \leq O(\log^{3}(n)) \cdot \mu_{KL} (\Lambda,K)$. 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 $O(n \log^{4}(n))$.

翻译：在Kannan与Lovász（1988）的开创性论文中，他们考虑了量µ_{KL}(Λ,K)，该量表示凸体K关于格Λ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) ≤ n·µ_{KL}(Λ,K)，而Dadush（2012）提出的子空间平坦性猜想声称O(log n)的因子就足够了，这将匹配Kannan和Lovász工作中给出的下界。我们通过证明µ(Λ,K) ≤ O(log^3(n))·µ_{KL}(Λ,K)将该猜想确定到指数上的常数因子内。我们的证明基于Regev与Stephens-Davidowitz（2017）的逆闵可夫斯基定理。沿用Dadush（2012, 2019）的工作，我们得到了一个时间复杂度为(log n)^{O(n)}的随机算法，用于求解n个变量的整数规划问题。我们主要结果的另一个推论是一个近最优的平坦常数，其值为O(n log^4(n))。