TY - JOUR
T1 - Unit distances in three dimensions
AU - Kaplan, Haim
AU - Matoušek, Jiří
AU - Safernová, Zuzana
AU - Sharir, Micha
PY - 2012/7
Y1 - 2012/7
N2 - We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].
AB - We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].
UR - http://www.scopus.com/inward/record.url?scp=84861960587&partnerID=8YFLogxK
U2 - 10.1017/S0963548312000144
DO - 10.1017/S0963548312000144
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84861960587
SN - 0963-5483
VL - 21
SP - 597
EP - 610
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 4
ER -