Curly Kakeya: The Hausdorff dimension of sets containing circles or line segments in many directions

2026/04/29 by Antonio Córdoba
#math.MG

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Abstract

In this paper, we establish a lower bound for the Hausdorff dimension of a set K⊂ ℝⁿ that contains a dilated and translated copy of every meridian, that is, of every (d-2)-dimensional sphere passing through the poles of S^d-1, a (d-1)-dimensional sphere, with 3≤ d≤ n, and S^d-1 contained in ℝⁿ. Under this assumption, we have dim_H(K)≥ d-1. This result is closely related to, and reminiscent of, the classical Kakeya set problem. We build upon this connection, employing similar techniques to show that if K⊂ ℝⁿ contains a unit straight line segment in every direction corresponding to a smooth curve on S^n-1, then its Hausdorff dimension is ≥ 2.

Citations

• Spectral Dimension Flow, Localized Non-Orientable Horizon Defects, and Emergent Time in Black Holes: A Geometry-Driven Effective Framework
• The Multiplier Problem for the Ball

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