Online Search for a Hyperplane in High-Dimensional Euclidean Space

TL;DR

This paper investigates the online search problem for hyperplanes in high-dimensional Euclidean space, establishing bounds on the shortest search curve length related to the dimension.

Contribution

It provides bounds on the minimal length of search curves needed to find hyperplanes in high-dimensional spaces, linking this to the problem's competitive ratio.

Findings

Shortest search curve length is at least proportional to the dimension d.

Shortest search curve length is at most proportional to d^{3/2}.

The bounds are within a constant factor of each other.

Abstract

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 $\Omega(d)\cap O(d^{3/2})$.