AI 中文总结
该研究确定了不同锐角区间内极小体积等角双曲4-多面体的类型，结合双曲几何相关定理完成证明，明确了单形退化及不同区间的多面体形态。
AI 中文摘要
我们研究二面角均等于固定严格锐角α的有限体积凸双曲4-多面体。令α₀=arccos(1/3)，α₁=arccos(1/4)。我们首先证明，当α<α₀时，该类多面体不存在；当α₀≤α<α₁时，极小体积的唯一多面体是二面角为α的正则双曲4-单形；当α增大至α₁时，该单形退化为欧氏单形，且在α₁处不存在双曲单形；当α₁≤α<π/2时，极小体积的唯一多面体是正则等角双曲4-立方体。证明结合了双曲Gram-Euler关系与Davis-Okun定理（针对3-球面的旗三角剖分的Charney-Davis不等式），几何步骤采用缺面论证，证明该类中所有非单形的对偶边界均为旗。
英文摘要
We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle $α$. Put $α_0=\arccos(1/3)$ and $α_1=\arccos(1/4)$. We first show that the class is empty for $α<α_0$. For $α_0\leqα<α_1$, the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle $α$. As $α$ increases to $α_1$, this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at $α_1$. For $α_1\leqα<π/2$, the unique minimum is the regular equiangular hyperbolic $4$-cube. The proof combines the hyperbolic Gram--Euler relation with the Davis--Okun theorem on the Charney--Davis inequality for flag triangulations of the $3$-sphere. The geometric step is a missing-face argument showing that the boundary of the dual of every nonsimplex in the class is flag.
