On the Number of Digital Discs
Journal of Mathematical Imaging and Vision
Broker-placement in latency-aware peer-to-peer networks
Computer Networks: The International Journal of Computer and Telecommunications Networking
On the number of digitizations of a disc depending on its position
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
Overlap properties of geometric expanders: extended abstract
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Betti numbers of polynomial hierarchical models for experimental designs
Annals of Mathematics and Artificial Intelligence
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A corner cut in dimension d is a finite subset of N"0^d that can be separated from its complement in N"0^d by an affine hyperplane disjoint from N"0^d. Corner cuts were first investigated by Onn and Sturmfels [Adv. Appl. Math. 23 (1999) 29-48], their original motivation stemmed from computational commutative algebra. Let us write N"0^dk"c"u"t for the set of corner cuts of cardinality k; in the computational geometer's terminology, these are the k-sets of N"0^d. Among other things, Onn and Sturmfels give an upper bound of O(k^2^d^(^d^-^1^)^/^(^d^+^1^)) for the size of N"0^dk"c"u"t when the dimension is fixed. In two dimensions, it is known (see [Corteel et al., Adv. Appl. Math. 23 (1) (1999) 49-53]) that #N"0^dk"c"u"t=@Q(klogk). We will see that in general, for any fixed dimension d, the order of magnitude of #N"0^dk"c"u"t is between k^d^-^1logk and (klogk)^d^-^1. (It has been communicated to me that the same bounds have been found independently by G. Remond.) In fact, the elements of N"0^dk"c"u"t correspond to the vertices of a certain polytope, and what our proof shows is that the above upper bound holds for the total number of flags of that polytope.