APDE seminar@BCAM: The Hausdorff dimension of sets containing circles or line segments in many directions.

Abstract

We establish a lower bound for the Hausdorff dimension of a set K ⊂ ℝⁿ that contains a translated copy of every meridian, that is, of every (d − 2)-dimensional sphere passing through the poles of 𝕊ᵈ⁻¹, a (d − 1)-dimensional sphere, with 3 ≤ d ≤ n, and 𝕊ᵈ⁻¹ ⊂ ℝⁿ.

Under this assumption, we have

dimₕ(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 ⊂ ℝᵈ contains a unit straight line segment in every direction corresponding to a smooth curve on 𝕊ᵈ⁻¹, then its Hausdorff dimension is ≥ 2.