Discrepancy of full-chord Buffon constructions
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How evenly can full chords of a compact convex body be distributed when their total prescribed length is L? The target uses discrepancy in the restricted full-chord model for bodies with finite piecewise C² boundary, with the disk also admitting a lower-bound question.
Why it matters and what counts as progress
Why it matters: The full-chord restriction changes which discrepancy constructions apply, leaving a gap between polylogarithmic upper and logarithmic lower behavior.
What counts as progress: Improve the O(L^(1/3)) construction toward polylogarithmic discrepancy or prove stronger unavoidable discrepancy for the disk.
How the problem developed
- 29 Mar 2026 · Problem formulation [1]
Stephan Steinerberger Introduced a Buffon discrepancy problem and proved an O(L^{1/3}) bound for a generalized construction on bounded convex domains; the full-chord restriction remained open.
- 21 May 2026 · Upper and lower bounds [2]
Korsky For full chords in a fixed convex body with finite piecewise C² boundary, proved an O((log L)^{3/2}) upper bound and an Ω(log L) lower bound for the disk.
AI contribution
Polylogarithmic Full-Chord Buffon Discrepancy
Korsky proves that every sufficiently large prescribed length L admits full-chord sets in each compact convex body with finite piecewise C² boundary and discrepancy O((log L)^(3/2)). For the disk, every full-chord set has discrepancy Ω(log L).
Sources
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