We study the problem of testing whether an unknown set $S$ in $n$ dimensions is convex or far from convex, using membership queries. The simplest high-dimensional discrete domain where the problem of testing convexity is non-trivial is the domain $\{-1,0,1\}^n$. Our main results are nearly-tight upper and lower bounds of $3^{\widetilde \Theta( \sqrt n)}$ for one-sided error testing of convex sets over this domain with non-adaptive queries. Together with our $3^{\Omega(n)}$ lower bound on one-sided error testing with samples, this shows that non-adaptive queries are significantly more powerful than samples for this problem.

翻译：我们研究利用成员查询来检测未知集合 $S$ 是否为凸集或远离凸集的问题，其中集合 $S$ 位于 $n$ 维空间中。在凸性检测非平凡的高维离散域中，最简单的域是 $\{-1,0,1\}^n$。我们的主要结果是在该域上使用非自适应查询进行单侧误差凸集检测的紧致上界和下界，均为 $3^{\widetilde \Theta( \sqrt n)}$。结合我们关于使用样本进行单侧误差检测的 $3^{\Omega(n)}$ 下界，这表明对于该问题，非自适应查询比样本具有显著更强的能力。