Online search for a hyperplane in high-dimensional Euclidean space

Shortest closed curve to inspect a sphere

“…Then γ and γ are linearly independent on U . So (2) shows that θ > 0 almost everywhere on U , via Cauchy-Schwarz inequality. Hence θ is monotone on U , which yields that γ is star-shaped with respect to o in a neighborhood of each point of U .…”

“…By absolute constant here we mean that C does not depend on n or γ. A Hamiltonian path in the edge graph of the cross polytope, i.e., the unit ball with respect to the L 1norm in R n , gives an example of a curve with L ≤ 2n √ nr [2]. Thus ( 5) is sharp up to the constant C. To establish (5), we may set r = 1.…”

“…This approach has implications for covering problems for the sphere by congruent disks [5], and yields a new proof of a result of Tikhomirov [18] (Note 5.4). There are many natural optimization problems for convex hull of space curves which remain open, including other questions of Zalgaller [20] which are closely related to well-known problems of Bellman [2][3][4] in operations research and search theory [1,11]; see also [12,13,15] and references therein. .…”

A Nearly Tight Lower Bound for the $d$-Dimensional Cow-Path Problem

“…Up to a constant factor in the optimal competitive ratio, using the standard doubling argument, one can assume without loss of generality that H has distance 1 from 0 [2]. Defining H to be the set of such hyperplanes, the competitive ratio of the cow path is simply the total distance traveled until the cow path has hit every H ∈ H .…”

“…In higher dimensions, even the asymptotic behavior of the optimal competitive ratio remains open. The best upper and lower bounds, due to Antoniadis et al [2], are O(d 3/2 ) and Ω(d), leaving a gap of roughly √ d. In this note, we settle the optimal d-dimensional competitive ratio up to low-order terms, presenting a simple argument for a lower bound of Ω(d 3/2 ), or, more precisely, Ω(d 3/2 / √ log d). Concurrent work by Ghomi and Wenk [9], posted on arXiv a few days before this note, establishes a d-dimensional competitive ratio of Ω(d 3/2 ) (communicated by Nazarov).…”