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.

翻译：我们研究了一些关于 convex 多边形的问题,这些圆形的脊椎存在于两套美元实际数(短期,\emph{grid}})的碳酸盐产品中。首先,我们证明每一个这样的网格都含有在 convex 位置上的$\Omega(\log n) 美元( log n) 点, 并且这个网格紧凑到一个不变的系数。 我们将这一结果概括为美元维度( 固定的 $x- in\mathb{N} 美元), 并且对于在 $- d$- omathbb{N} 上方格中的最大点, 得到一小于$\ Omega( log ⁇ d- 1}n) 的紧紧紧的底线, 也就是在 $- 或 $y- monnonoone 的网格中的最大点, 我们提出多米时间算算算法, 在一个没有两点的网格中, 以 相同的 $- 或 $y$- coyo- councoun coconto coun coun coun coun coun counticle coun col 。我们提出在这种硬的极限的极限的矩定数组定在这种硬点上, 。