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

VL - 68

SP - 448

EP - 476

JO - Discrete & Computational Geometry

JF - Discrete & Computational Geometry

SN - 0179-5376

IS - 2

ER -