多面体和R^n的锐三角剖分
Acute triangulations of polyhedra and R^n
摘要：
研究了凸多面体和空间R^n的锐三角剖分问题。这里，锐角三角剖分是将二面角为锐角的单形进行三角剖分。我们证明了当n>=4时，n-立方体的锐角三角剖分不存在。进一步，我们证明了当n>= 5时，空间R^n的锐角三角剖分不存在。相反，在R^3中，我们给出了立方体、正八面体和正四面体的非平凡锐角三角剖分的构造。我们还证明了如果所有的二面角都有界远离π/2，则R^4的锐角三角剖分不存在。
英文摘要：
We study the problem of acute triangulations of convex polyhedra and the space R^n. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n>=4. Further, we prove that acute triangulations of the space R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of R^4 if all dihedral angles are bounded away from pi/2.
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