D-representability of Simplicial Complexes of Fixed Dimension
D-representability of Simplicial Complexes of Fixed Dimension
复制标题
定维单纯复形的D-表示性
DOI：
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发表时间：
2011
影响因子：
0.3
通讯作者：
M. Tancer
中科院分区：
文献类型：
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作者：
M. Tancer
设K是顶点集为V = {v_1，...，v_n}。复形K是d-可表示的，如果存在集合{C_1，.，C_n}，使得子集合{C_{i_1}，...，C_{i_j}}有非空交集当且仅当{v_{i_1}，.，v_{i_j}}是K的一个面。
1967年，魏格纳证明了每一个维数为d的单纯复形都是（2d+1）-可表示的。他还认为，他的界限是最好的，即，存在$d$维单纯复形，它们不可2d表示。然而，他无法证明他的建议。
我们证明他的建议确实是对的。这样，我们就为欧氏空间中凸集的交模式之谜又添了一块拼图。
Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if and only if {v_{i_1},...,v_{i_j}} is a face of K.
In 1967 Wegner proved that every simplicial complex of dimension d is (2d+1)-representable. He also suggested that his bound is the best possible, i.e., that there are $d$-dimensional simplicial complexes which are not 2d-representable. However, he was not able to prove his suggestion.
We prove that his suggestion was indeed right. Thus we add another piece to the puzzle of intersection patterns of convex sets in Euclidean space.