Center-flat depth conjecture

Prove that every finite point set P in Euclidean d-space is split by at least one k-dimensional flat with depth at least (k+1)/(k+d+1), for every integer k satisfying 0≤k≤d−1.

Background

The paper defines a finite point set as α-split by a k-flat when every hyperplane containing that flat leaves at least an α-fraction of the points in each closed half-space. For k=0, Rado’s centerpoint theorem guarantees a point of depth at least 1/(d+1).

The authors state a broader conjecture asserting the existence of a k-flat of depth (k+1)/(k+d+1) for every finite point set and every 0≤k≤d−1. They note that partial results were known, but the full assertion is unresolved.

References

Furthermore, it is conjectured that every point set $P$ is $\left(\frac{k+1}{k+d+1}\right)$-split by at least one $k$-flat, for $0\leq k\leq d-1$ ; recent partial results in this direction have been reported by Magazinov and P\or .

— Helly Type Theorems for Splitting Point-Sets  (2609.02180 - Portal et al., 2 Sep 2026) in Section 1, Introduction, subsection “α-splitting point sets with flats”