内接多面体与Delaunay三角剖分的普适性定理
Universality theorems for inscribed polytopes and Delaunay triangulations
摘要：
我们证明了每一个准素基本半代数集同伦等价于一个多面体的内接实现集（直到莫比乌斯变换）。如果半代数集是开放的，那么，此外，我们证明了（同伦），它是一个收缩的实现空间的一些内接邻居（和单纯）多面体。我们还表明，所有的代数扩张$\mathbb{Q}$需要协调内接多面体。这些陈述表明内接多面体表现出Mnëv普适性现象。通过球极平面投影，这些定理直接转化为Delaunay细分的普适性定理。特别是，我们的结果意味着Delaunay三角剖分的可实现性问题是多项式等价的实数的存在性理论。
英文摘要：
We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, our results imply that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.
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