### Citations

1324 |
Geometric Measure Theory
- Federer
- 1969
(Show Context)
Citation Context ...ary of d-dimensional sets by the (d − 1)dimensional Hausdorff measure, and denote it slightly sloppy by |∂A|. For a definition of the Hausdorff measure and related definitions, we refer the reader to =-=[7]-=-. The isoperimetric quotient of A ⊂ R d is defined to be |∂A| d /|A| d−1 . The isoperimetric quotient can be considered as a certain measure of the fatness of a figure A. We always assume shapes to be... |

197 |
Combinatorial methods in density estimation.
- Devroye, Lugosi
- 2001
(Show Context)
Citation Context ...large. The analysis of the algorithm is based on these simple ideas whose details are not that easy. They are hidden in the following longish theorem that follows from theorems in Chapters 3 and 4 of =-=[5]-=-. Theorem 3 (Probabilistic toolbox) Let Ω ⊆ Rk be a metric space, and let B be the set of balls of some fixed radius δ > 0 in Ω. Let vol be a measure on Ω such that, for all x ∈ Ω, the volume vol(B(x,... |

138 |
Computational geometry
- Berg, Kreveld, et al.
- 1997
(Show Context)
Citation Context ...maximal number of “votes”, which can be computed by brute force in time O(N 2 ) in any dimension. The time bound can be further improved to O(N(log N) d−1 ) by using orthogonal range counting queries =-=[4]-=-. Cheong et al. [3] introduce a general probabilistic framework, which they use for approximating the maximal area of overlap of two unions of n and m triangles in the plane, with prespecified absolut... |

26 | On finding a guard that sees most and a shop that sells most, 2005. Accepted for publication in Discrete Comput. Geom. and available as http://tclab.kaist.ac.kr/ ∼ otfried/Papers/ceh-fgsmssm.pdf. A preliminary version appeared
- Cheong, Efrat, et al.
- 2004
(Show Context)
Citation Context ...votes”, which can be computed by brute force in time O(N 2 ) in any dimension. The time bound can be further improved to O(N(log N) d−1 ) by using orthogonal range counting queries [4]. Cheong et al. =-=[3]-=- introduce a general probabilistic framework, which they use for approximating the maximal area of overlap of two unions of n and m triangles in the plane, with prespecified absolute error ε, in time ... |

25 | A (slightly) faster algorithm for Klee’s measure problem
- Chan
(Show Context)
Citation Context ...ed the maximal number of “votes” was computed, which boiled down to computing a deepest cell of an arrangement of boxes. For N boxes, the best known time bound is O(N d/2 /(log N) d/2−1 polyloglog N) =-=[2]-=-. Here we show that it is sufficient to output the “vote” whose neighborhood contains the maximal number of “votes”, which can be computed by brute force in time O(N 2 ) in any dimension. The time bou... |

18 | On the area of overlap of translated polygons.
- Mount, Silverman, et al.
- 1996
(Show Context)
Citation Context ...alculation error in the final derivation of the time bound, as was noted in [11]. Their algorithm works with high probability. For two simple polygons with n and m vertices in the plane, Mount et al. =-=[8]-=- show that a translation that maximizes the area of overlap can be computed in time O(n 2 m 2 ). For maximizing the volume of overlap of two unions of simplices under rigid motions, no exact algorithm... |

9 | Geometric optimization and sums of algebraic functions,” in
- Vigneron
- 2010
(Show Context)
Citation Context ...ions and in time O(m + (n 3 /ε 8 )(log n) 5 ) for rigid motions. The latter time bound is smaller in their paper, due to a calculation error in the final derivation of the time bound, as was noted in =-=[11]-=-. Their algorithm works with high probability. For two simple polygons with n and m vertices in the plane, Mount et al. [8] show that a translation that maximizes the area of overlap can be computed i... |

5 |
Generation of random orthogonal matrices
- HEIBERGER
- 1978
(Show Context)
Citation Context ...fter many rounds, say N, we determine the best cluster in the space of rigid motions. There are many different methods to compute a random rotation matrix described in the literature; see for example =-=[6, 10]-=-. To define the uniform distribution formally, a volume has to be defined. The group SO(d) is ( ) d 2 -dimensional. The volume | · | in SO(d) is measured by the ( ) d 2 -dimensional Haar measure. For ... |

4 | Probabilistic matching of planar regions
- Alt, Scharf, et al.
(Show Context)
Citation Context ...e volume (Lebesgue measure) of a set. We use ε|A| and not just ε as error bound because the inequality should be invariant under scaling of both shapes with the same factor. In a previous publication =-=[1]-=- we considered the 2dimensional case. Here we not only generalize the results to higher dimensions, but we also give new proofs that improve the bounds on the number of random samples. Furthermore we ... |

4 |
A remark on AS127. Generation of random orthogonal matrices
- Tanner, Thisted
- 1982
(Show Context)
Citation Context ...fter many rounds, say N, we determine the best cluster in the space of rigid motions. There are many different methods to compute a random rotation matrix described in the literature; see for example =-=[6, 10]-=-. To define the uniform distribution formally, a volume has to be defined. The group SO(d) is ( ) d 2 -dimensional. The volume | · | in SO(d) is measured by the ( ) d 2 -dimensional Haar measure. For ... |

2 | An upper bound on the volume of the symmetric difference of a body and a congruent copy. Submitted. Preprint at http://arxiv.org/abs/1010.2446
- Schymura
- 2010
(Show Context)
Citation Context ...te the k-dimensional Hausdorff measure. For Lebesgue measurable sets in Rd , the d-dimensional Hausdorff measure and the Lebesgue measure coincide. The symmetric difference is denoted by △. Theorem 5 =-=[9]-=- Let A ⊂ R d be bounded. Let t ∈ R d be a translation vector. Then H d (A △ (A + t)) ≤ |t| H d−1 (∂A). This implies that the density function f is Lipschitz continuous. Applying the Cauchy-Schwarz ine... |