Follow your favourite podcasts, listen offline and in the car with CarPlay and Android Auto, and always pick up where you left off. Free to try.
The Subspace Flatness ConjectureThe covering radius $\mu(\Lambda,K)$ of a convex body $K \subseteq \setR^n$ with respect to a full rank lattice $\Lambda \subseteq \setR^n$ is the minimum scaling needed so that translates of $K$ placed at every lattice point cover the whole $\setR^n$. A question going back to work of Kannan and Lov\'asz (1988) is how well the covering radius is described solely by the volume of $K$ and the density of the lattice, after projecting both into a suitable subspace. We report on significant progress on this question due to Regev and Stephens-Davidowitz (2016) who resolve this problem for ellipsoids and Reis and Rothvoss (2023) who cover the general case. We also survey applications of these results that give near-optimal bounds on the flatness constant as well as a $(\log n)^{O(n)}$-time algorithm for solving $n$-variable integer programs, following work of Dadush (2012). This is based on joint work with Victor Reis.Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices
