We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the 1-skeleton. The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lower-dimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron.

中文翻译：

---

双曲多面体的最大体积

我们使用一些固定组合和任何类型的顶点（真实的、理想的或超理想的）来研究双曲多面体体积的上限值。我们发现 supremum 总是等于 1 骨架的整流体积。该定理是通过将一种体积增加的流应用于任何双曲多面体来证明的。由于多面体的某些层可能退化为较低维的物体，因此流动中可能会出现奇点；当这种情况发生时，我们需要仔细研究所产生的多面体的组合并继续流动，直到最终我们得到一个修正的多面体。