We revisit the problem of property testing for convex position for point sets in $\mathbb{R}^d$. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (ESA 2000). First, the algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i)~exhibiting both negative and positive certificates along with the convexity determination, and (ii)~significantly extending the input range for moderate and higher dimensions. The behavior of the randomized tester is as follows: (i)~if $P$ is in convex position, it accepts; (ii)~if $P$ is far from convex position, with probability at least $2/3$, it rejects and outputs a $(d+2)$-point witness of non-convexity as a negative certificate; (iiii)~if $P$ is close to convex position, with probability at least $2/3$, it accepts and outputs an approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every dimension $d$ (and is faster in low dimensions).

翻译：我们重新审视了 $\mathbb{R}^d$ 中点集凸性位置的性质检验问题。我们的结果借鉴了 Czumaj、Sohler 和 Ziegler（ESA 2000）的先前思路。首先，对算法进行了重新设计，并修正了其正确性分析。其次，通过以下方式扩展了其功能：（i）在凸性判定中同时提供否定性认证和肯定性认证；（ii）显著扩展了中等及高维度的输入范围。该随机化测试器的行为如下：（i）若点集 $P$ 处于凸性位置，则接受；（ii）若 $P$ 远离凸性位置，则以至少 $2/3$ 的概率拒绝，并输出一个 $(d+2)$ 点的非凸性证据作为否定性认证；（iii）若 $P$ 接近凸性位置，则以至少 $2/3$ 的概率接受，并输出凸性位置最大子集的近似值。该算法对每个维度 $d$ 仅检测亚线性数量的点，并以亚二次时间运行（在低维度下速度更快）。