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

In this paper, we establish a lower bound for the Hausdorff dimension of a set $K\subset {\mathbb{R}}^n$ 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 $3\le d\le n$, and $S^{d-1}$ contained in ${\mathbb{R}}^n$. Under this assumption, we have $${\dim }_H(K)\ge d-1.$$ 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 $K\subset {\mathbb{R}}^n$ contains a unit straight line segment in every direction corresponding to a smooth curve on $S^{n-1}$, then its Hausdorff dimension is $\ge 2$.