# Convex Polygons in Cartesian Products

## TL;DR

This paper investigates the properties of convex polygons within Cartesian product point sets, establishing bounds on their size, providing algorithms for specific convex chains, and analyzing the maximum number of convex point sets in grids.

## Contribution

It proves tight bounds on convex polygons in grids, extends results to higher dimensions, and offers efficient algorithms and bounds for convex chains and point sets.

## Key findings

- Every 2D grid contains Ω(log n) convex points, tight up to a constant.
- Algorithms for longest monotone convex chains in grids are polynomial-time.
- Exponential bounds on the maximum number of convex point sets in grids.

## Abstract

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of $n$ real numbers (for short, \emph{grid}). First, we prove that every such grid contains $\Omega(\log n)$ points in convex position and that this bound is tight up to a constant factor. We generalize this result to $d$ dimensions (for a fixed $d\in \mathbb{N}$), and obtain a tight lower bound of $\Omega(\log^{d-1}n)$ for the maximum number of points in convex position in a $d$-dimensional grid. Second, we present polynomial-time algorithms for computing the longest $x$- or $y$-monotone convex polygonal chain in a grid that contains no two points with the same $x$- or $y$-coordinate. We show that the maximum size of a convex polygon with such unique coordinates can be efficiently approximated up to a factor of $2$. Finally, we present exponential bounds on the maximum number of point sets in convex position in such grids, and for some restricted variants. These bounds are tight up to polynomial factors.