Covering a set with homothets of a convex body
Covering a set with homothets of a convex body
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用凸体的同调覆盖集合
DOI：
10.1007/s11117-009-0005-8
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发表时间：
2010
期刊：
影响因子：
1
通讯作者：
M. Naszódi
中科院分区：
文献类型：
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作者：
M. Naszódi
我们考虑了《离散几何中的研究问题》(Brass et al.《离散几何中的研究问题》，第十二卷+499。斯普林格，纽约，ISBN；0-387-23815-8；0-387-23815-8。首先，设K和L在$${\mathbb{R}^{d}}$$中有凸体。我们证明了如果K的正同伦族的总体积足够大，则它们允许L的平移覆盖。这个问题最初是由L.Fejes Tóth提出的，当K=L且维度为2时。以前已知的界限(Januszewski in Proc.国际数学科学会议，第29-34页。ŽIlina，1998)关于总体积为ddVol(K)阶的情形，我们证明了一个在维度上指数的上界。第二个问题是：找出一个关于同伦系数的条件，这个条件是K的一个正同伦族覆盖K的必要条件。这个问题是由V.Soltan提出的，他猜想系数的和至少是d。我们确认这个猜想的一个渐近版本。
We consider two problems mentioned in the book “Research Problems in Discrete Geometry” (Brass et al. in research problems in discrete geometry, vol xii+499. Springer, New York, pp ISBN: 978-0387-23815-8; 0-387-23815-8, 2005). First, let K and L be given convex bodies in $${\mathbb{R}^{d}}$$ . We prove that if the total volume of a family of positive homothets of K is sufficiently large then they permit a translative covering of L. This problem, in the case when K = L and the dimension is two, was originally posed by L. Fejes Tóth. The previously known bound (Januszewski in proc. of the International scientific conference on mathematics, pp 29–34. Žilina, 1998) on the total volume (in the case when K = L) was of order dd vol(K), we prove a bound that is exponential in the dimension. The second problem is the following: Find a condition, in terms of the coefficients of homothety, that is necessary for a family of positive homothets of K to cover K. The problem was phrased by V. Soltan, who conjectured that the sum of the coefficients is at least d. We confirm an asymptotic version of this conjecture.