The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Slicing Convex Sets and Measures by a Hyperplane
Discrete & Computational Geometry
Discrete & Computational Geometry
| Hi-index | 0.89 |
Barany et al. (2008) [1] proved that, for any @b@?[0,1]^d and any d well-separated convex bodies S"1,S"2,...,S"d in R^d, there exists a hyperplane (a generalized ham-sandwich cut) splitting the volume of S"i as (@b"i,1-@b"i) for all i. Steiger and Zhao (2010) [4] proved a discrete analogue for n points in weak general position. The (elegant!) proof inspired an algorithm for computing this hyperplane (Steiger and Zhao, 2010 [4]) with running time O(nlog^d^-^3n) for d=3 and O(n) for d=2. In this note we show that their algorithm can be modified to compute a generalized ham-sandwich cut in linear time for any fixed d.