Large convexly independent subsets of Minkowski sums
KJ Swanepoel, P Valtr - arXiv preprint arXiv:1007.2775, 2010 - arxiv.org
KJ Swanepoel, P Valtr
arXiv preprint arXiv:1007.2775, 2010•arxiv.orgLet $ E_d (n) $ be the maximum number of pairs that can be selected from a set of $ n $
points in $ R^ d $ such that the midpoints of these pairs are convexly independent. We show
that $ E_2 (n)\geq\Omega (n\sqrt {\log n}) $, which answers a question of Eisenbrand, Pach,
Rothvo\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of
finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also
show that $\lfloor\frac {1}{3} n^ 2\rfloor\leq E_3 (n)\leq\frac {3}{8} n^ 2+ O (n^{3/2}) $. Let …
points in $ R^ d $ such that the midpoints of these pairs are convexly independent. We show
that $ E_2 (n)\geq\Omega (n\sqrt {\log n}) $, which answers a question of Eisenbrand, Pach,
Rothvo\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of
finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also
show that $\lfloor\frac {1}{3} n^ 2\rfloor\leq E_3 (n)\leq\frac {3}{8} n^ 2+ O (n^{3/2}) $. Let …
Let be the maximum number of pairs that can be selected from a set of points in such that the midpoints of these pairs are convexly independent. We show that , which answers a question of Eisenbrand, Pach, Rothvo\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that . Let be the maximum number of pairwise nonparallel unit distance pairs in a set of points in some -dimensional strictly convex normed space. We show that and for that , where is related to strictly antipodal families. In fact we show that the same asymptotics hold without the requirement that the unit distance pairs form pairwise nonparallel segments, and also if diameter pairs are considered instead of unit distance pairs.
arxiv.org
Showing the best result for this search. See all results