We consider the online search problem in which a server starting at the origin of a $d$-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and the length of the shortest curve from which each point on the $d$-dimensional unit sphere can be seen are within a constant factor of each other. We show that this length is in $\Omega(d)\cap O(d^{3/2})$.

中文翻译：

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在线搜索高维欧几里得空间中的超平面

我们考虑在线搜索问题，其中从 $d$ 维欧几里得空间的原点开始的服务器必须找到任意超平面。最佳可能的竞争比率和可以看到 $d$ 维单位球面上每个点的最短曲线的长度在彼此的常数因子内。我们证明这个长度在 $\Omega(d)\cap O(d^{3/2})$ 中。