We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, 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. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every \(n\ge d\). 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.

我们探索了非空心晶格多面体覆盖半径的上限。特别是，我们推测维度 d中d /2的一般上限，通过“标准终端单纯形”和它们的直接和来实现。我们将这个猜想证明到三维，并证明它等同于 González-Merino 和 Schymura 的猜想（离散计算。Geom. 58 (3), 663–685 (2017)），即d -th 覆盖最小值标准终端n -simplex 等于d /2，对于每个\(n\ge d\). 我们还表明，这两个猜想将来自 Hadwiger 公式的晶格单纯形的离散模拟，该公式根据表面积与体积的比率限制凸体的覆盖半径。为此，我们引入了非空心单纯形的离散表面积的新概念。我们在二维中证明了我们的离散类比，并为其在任意维度上的有效性提供了强有力的证据。