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Keleti's line segment extension conjecture (general dimension)

Establish that for any set L of line segments in Euclidean space R^n, the Hausdorff dimension of the union of their full lines equals the Hausdorff dimension of the union of the segments themselves, i.e., prove dim(⋃_{ℓ∈L} \tilde ℓ) = dim(⋃_{ℓ∈L} ℓ) for all dimensions n.

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Background

The conjecture is stated in full generality and is known to follow from the Kakeya set conjecture in the corresponding dimension. The authors use their resolution of Kakeya in R3 to deduce the extension conjecture in three dimensions, but the general case remains open pending resolution of Kakeya in higher dimensions.

A solution would have implications for geometric measure theory and fine properties of dimensions under geometric transformations.

References

\begin{conj}\label{lineExtension} Let $L$ be a set of line segments in $Rn$. Then

dim \Big(\bigcup_{\ell \in L}\tilde\ell\phantom{.}\Big) = \dim \Big(\bigcup_{\ell\in L}\ell\Big). \end{conj}

— Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions (2502.17655 - Wang et al., 24 Feb 2025) in Section "Tube doubling and Keleti's line segment extension conjecture"