Higher-dimensional visible part conjecture

Prove that for any compact set E ⊂ ℝ^n, the Hausdorff dimension of the visible part Vis_θ(E) equals min{dim_H(E), n−1} for Lebesgue-almost every direction θ ∈ S^{n−1}.

Background

In ℝn, the conjectured typical dimension of visible parts generalizes the planar conjecture: the visible portion should have dimension min{dim_H(E), n−1} for almost all directions. While analogues of Marstrand’s theorem provide lower bounds and recent work has improved upper bounds for typical directions, the exact almost-sure value remains open in higher dimensions.

The survey highlights recent progress, including general upper bounds due to Orponen and further improvements by Dąbrowski, but indicates that the full conjecture remains unresolved.

References

The analogous conjecture in higher dimensions, that the dimension of the visible part of a compact $E\subset \mathbb{R}n$ equals $\min{ E, n-1}$, is also unresolved, but Orponen showed that $ \mbox{Vis}\theta E \leq n -1/50n$ for almost all $\theta$ for all $n\geq 2$, and very recently D\c{a}browski improved this to $ \mbox{Vis}\theta E \leq n -1/6$, leading to improved bounds for Ahlfors regular sets.

— Seventy Years of Fractal Projections  (2602.22002 - Falconer, 25 Feb 2026) in Section 8 (Some other aspects of fractal projections) — Visible parts of sets