Tightness of the dual Radon number upper bound for extremal classes

Determine whether the upper bound r(C) ≤ 2·vc(C)+1 for extremal concept classes C is tight for all VC dimensions; in particular, identify whether for each integer d ≥ 1 there exists an extremal class with vc(C)=d and dual Radon number r(C)=2d+1.

Background

The authors introduce the convexity space associated with a concept class and define the dual Radon number r(C). Their main technical result establishes r(C) ≤ 2·vc(C)+1 for extremal classes, with lower bounds and examples suggesting near-tightness up to constants.

They explicitly state that determining the exact tightness of this upper bound remains open, despite examples showing tightness within a factor of two and exact tightness for special cases (e.g., hyperplane arrangements).