TY - JOUR
T1 - On Weighted Sums of Numbers of Convex Polygons in Point Sets
AU - Huemer, Clemens
AU - Oliveros, Deborah
AU - Pérez-Lantero, Pablo
AU - Torra, Ferran
AU - Vogtenhuber, Birgit
PY - 2022
Y1 - 2022
N2 - Let S be a set of n points in general position in the plane, and let Xk,ℓ(S) be the number of convex k-gons with vertices in S that have exactly ℓ points of S in their interior. We prove several equalities for the numbers Xk,ℓ(S). This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.
AB - Let S be a set of n points in general position in the plane, and let Xk,ℓ(S) be the number of convex k-gons with vertices in S that have exactly ℓ points of S in their interior. We prove several equalities for the numbers Xk,ℓ(S). This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.
U2 - 10.1007/s00454-022-00395-8
DO - 10.1007/s00454-022-00395-8
M3 - Article
SN - 0179-5376
VL - 68
SP - 448
EP - 476
JO - Discrete & Computational Geometry
JF - Discrete & Computational Geometry
IS - 2
ER -