Codenotti Giulia, Santos Francisco, Schymura Matthias
Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany.
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Av. de Los Castros 48, 39005 Santander, Spain.
Discrete Comput Geom. 2022;67(1):65-111. doi: 10.1007/s00454-021-00330-3. Epub 2021 Nov 17.
We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of /2 in dimension , achieved by the "standard terminal simplices" and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. (3), 663-685 (2017)) that the -th covering minimum of the standard terminal -simplex equals /2, for every . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger's formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.
我们探索非空心格点多面体覆盖半径的上界。特别地，我们猜想在(n)维中一般上界为(n/2)，由“标准终端单纯形”及其直和达到。我们将这个猜想证明到三维，并表明它等同于冈萨雷斯 - 梅里诺和施穆拉的猜想（《离散计算几何》(3)，663 - 685 (2017)），即对于每个(n)，标准终端(n) - 单纯形的第(n)个覆盖最小值等于(n/2)。我们还表明，这两个猜想将由哈迪格公式的格点单纯形离散类似物推出，该公式根据表面积与体积的比率来界定凸体的覆盖半径。为此，我们引入了非空心单纯形离散表面积的新概念。我们在二维中证明了我们的离散类似物，并给出了其在任意维度有效性的有力证据。
