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 $d$.

翻译：我们重新审视了$\mathbb{R}^d$中点集凸位置的属性测试问题。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，并对其分析进行了修正以确保正确性。其次，通过以下方式扩展了其功能：(i)在凸性判定中同时展示负证书和正证书，以及(ii)显著扩展了中等维度与高维度的输入范围。该随机测试器的行为如下：(i)若$P$处于凸位置，则接受；(ii)若$P$远离凸位置，则以至少$2/3$的概率拒绝，并输出一个$(d+2)$点非凸性见证作为负证书；(iii)若$P$接近凸位置，则以至少$2/3$的概率接受，并输出最大凸位置子集的近似。该算法检查的子线性点数，且对于任意维度$d$，运行时间为次二次时间。