The lattice size of a lattice polytope P was defined and studied by Schicho, and Castryck and Cools. They provided an “onion skins” algorithm for computing the lattice size of a lattice polygon P in \(\mathbb R^2\) based on passing successively to the convex hull of the interior lattice points of P. We explain the connection of the lattice size to the successive minima of \(K=(P+(-P))^*\) and to the lattice reduction with respect to the general norm that corresponds to K. It follows that the generalized Gauss algorithm of Kaib and Schnorr (which is faster than the “onion skins” algorithm) computes the lattice size of any convex body in \(\mathbb R^2\). We extend the work of Kaib and Schnorr to dimension three, providing a fast algorithm for lattice reduction with respect to the general norm defined by a convex origin-symmetric body \(K\subset \mathbb R^3\). We also explain how to recover the successive minima of K and the lattice size of P from the obtained reduced basis and therefore provide a fast algorithm for computing the lattice size of any convex body \(P\subset \mathbb R^3\).

Schicho，Castryck和Cools定义并研究了晶格多角体P的晶格大小。它们用于计算的晶格尺寸提供一个“洋葱皮”算法的晶格多边形P在\（\ mathbb R ^ 2 \）基于依次传递给内部格点的凸包P。我们解释了晶格大小与\（K =（P +（-P））^ * \）的连续最小值以及与对应于K的一般范数的晶格约简的关系。因此，Kaib和Schnorr的广义高斯算法（比“洋葱皮”算法要快）可以计算\（\ mathbb R ^ 2 \）中任何凸体的晶格大小。我们将Kaib和Schnorr的工作扩展到三维，针对相对于凸原点对称体\（K \ subset \ mathbb R ^ 3 \）定义的一般范式，提供了一种用于晶格归约的快速算法。我们还解释了如何从获得的缩减基中恢复K的连续最小值和P的晶格大小，从而为计算任何凸体\（P \ subset \ mathbb R ^ 3 \）的晶格大小提供了一种快速算法。