Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.18362

A new unconditional lower bound for shoreline search

Alexander Temerev
View original source ↗Read source document ↗Export metadata ↓

Abstract

A unit-speed searcher starts at the origin of the Euclidean plane and must hit an unknown straight line whose direction and distance from the origin are both unknown. We prove that every deterministic search path has competitive ratio at least $C_{\log}\approx 12.5937096701246675$. The bound is unconditional: the path need not be cyclic, self-similar, spiral-like, or monotone in angle. For each projection direction, we compare the path with a zigzag obtained by sorting its alternating record turns. The resulting completion constraints are interpreted as jobs with scale-dependent deadlines and lower-bounded through a finite-window scheduling argument in logarithmic time. Averaging these directional bounds then uses the exact Euclidean velocity budget. In one dimension, the same method recovers the optimal cow-path constant $9$. Finally, Arb ball arithmetic provides a rigorous numerical enclosure of the constant.

Explore related subjects

math.MGmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander Temerev. 2026-08-18. A new unconditional lower bound for shoreline search. https://arxiv.org/abs/2608.18362

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$β$-Uniform Convexity and Divisible Domains

Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry it induces is the hyperbolic space. In general, strictly convex divisible domains exhibit negative curvature properties, but only the ellipsoid is a CAT(0) space. The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-Ichi Ohta as a generalization of the Alexandrov-Toponogov comparison theorems to Finsler manifolds. We prove that a natural Finsler metric on a strictly convex divisible domain is $β$-uniformly convex, where the constant $β$ is related to the regularity of the boundary. We use this to show, with AI assistance, that the Hilbert metric, under suitable local and scale-dependent assumptions, is $β$-uniformly convex on such domains.

math.MG↗
arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Minimal central slices of the regular simplex

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

math.MG↗