双侧凸性测试与证书
Two-sided convexity testing with certificates
AI总结：
本文重新设计了凸位置性质测试算法，修正正确性，并扩展功能以输出正负证书，适用于更高维度，算法检查次线性点并以次二次时间运行。
摘要：
我们重新研究了$\mathbb{R}^d$中点集凸位置的性质测试问题。我们的结果借鉴了Czumaj、Sohler和Ziegler（ESA 2000）的先前思想。首先，算法被重新设计，其分析被修正以确保正确性。其次，其功能得到扩展：（i）在凸性判定时同时给出负证书和正证书，（ii）显著扩展了中等维度和高维度的输入范围。随机测试器的行为如下：（i）如果$P$处于凸位置，则接受；（ii）如果$P$远离凸位置，则以至少$2/3$的概率拒绝，并输出一个$(d+2)$点的非凸性见证作为负证书；（iii）如果$P$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。该算法检查次线性数量的点，并在每个维度$d$下以次二次时间运行。
英文摘要：
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 $d$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
