Many triangulated odd-dimensional spheres
Many triangulated odd-dimensional spheres
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许多奇数维三角球体
DOI：
10.1007/s00208-015-1232-x
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发表时间：
2014
影响因子：
1.4
通讯作者：
Stedman Wilson
中科院分区：
文献类型：
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作者：
Eran Nevo;F. Santos;Stedman Wilson
已知$$(2k-1)$$(2k-1)-球面至多有$$2^{O(n^k\logn)}$$2O(Nklogn)个n点组合不同的三角剖分，对每个$$k\ge2$$k≥2，这里我们构造至少$$2^{\ommega(n^k)}$2Ω(Nk)个这样的三角剖分，改进了以前的构造，在一般情况下(KALAI)给出$$2^\Omega(n^{k-1)}$$2Ω(nk-1)，对于$$k=2$$k=2(Pfeifle-Ziegler)给出$2^\Omega(n^{5/4})$2Ω(N5/4)。我们还构造了$$2^{\Omega(n^{k-1+\FRAC{1}{k}})}$$2Ω(nk-1+1k)测地线(又称.$$(2k-1)$$(2k-1)球面的星凸)n点三角剖分。作为这方面的一个步骤(在$$k=2$$k=2的情况下)，我们构造了包含$$\omega(n^{3/2})$$Ω(n3/2)面的n-顶点4-多面体，这些面不是单面，或者具有$$\omega(n^{3/2})$$Ω(n3/2)条三次边。
It is known that the $$(2k-1)$$(2k-1)-sphere has at most $$2^{O(n^k \log n)}$$2O(nklogn) combinatorially distinct triangulations with n vertices, for every $$k\ge 2$$k≥2. Here we construct at least $$2^{\Omega (n^k)}$$2Ω(nk) such triangulations, improving on the previous constructions which gave $$2^{\Omega (n^{k-1})}$$2Ω(nk-1) in the general case (Kalai) and $$2^{\Omega (n^{5/4})}$$2Ω(n5/4) for $$k=2$$k=2 (Pfeifle–Ziegler). We also construct $$2^{\Omega (n^{k-1+\frac{1}{k}})}$$2Ω(nk-1+1k) geodesic (a.k.a. star-convex) n-vertex triangulations of the $$(2k-1)$$(2k-1)-sphere. As a step for this (in the case $$k=2$$k=2) we construct n-vertex 4-polytopes containing $$\Omega (n^{3/2})$$Ω(n3/2) facets that are not simplices, or with $$\Omega (n^{3/2})$$Ω(n3/2) edges of degree three.