On the Volume of Lattice Polyhedra
On the Volume of Lattice Polyhedra
复制标题
论格子多面体的体积
DOI：
10.1112/plms/s3-7.1.378
复制
发表时间：
1957
影响因子：
1.8
通讯作者：
J. Reeve
中科院分区：
文献类型：
--
作者：
J. Reeve
Mr)= l (y)-¥(?)-h（!）其中l (y)和l (y)表示I中分别属于y及其边界y的点的个数，假设基本平行四边形为单位面积。在这篇笔记中，我们讨论了(1)的某些推广，特别地，我们在定理II中得到了一个公式，它在许多方面类似于上面的公式，用于三维欧几里德空间Ea中特定类型的多面体的体积。我们的公式有效的多面体类，作为一个特例，包括在E3中某些笛卡尔坐标系中顶点具有积分坐标的凸多面体类。凸多面体的公式在定理一中有明确的说明。下面的例子清楚地说明，这个公式所包含的东西，一定比人们最初可能预料到的(1)的直接推广更多。
Mr)= l (y)-¥(?)-h(!) where l (y) and l (y) denote the number of points of I which belong respectively to y and its boundary y, provided the fundamental parallelogram is of unit area.In this note we discuss certain generalizations of (1) and, in particular, we obtain in Theorem II a formula, which is in many ways analogous to the one above, for the volume of a polyhedron of a particular type in threedimensional euclidean space Ea. The class of polyhedra for which our formula is valid includes as a special case the class of convex polyhedra whose vertices have integral coordinates in some cartesian coordinate system in E3. The formula for convex polyhedra is stated explicitly in Theorem I. That this formula must embody something more than the direct extension of (1) which one might at first anticipate is clearly illustrated by the following example.