Asymptotic determination of the weakly convex-position threshold

Determine whether the quantity C_d(l,n), defined as the least N such that every N-point set in weakly convex position in \mathbb{R}^d contains either l points in a common (d−1)-dimensional hyperplane or n points in convex position, satisfies C_d(l,n) \approx nl/d.

Background

The paper introduces C_d(l,n) as an auxiliary extremal quantity intended to improve the upper bound for ES_d(l,n). It concerns point sets whose points all lie on the boundary of their convex hull, without requiring general position. The authors conjecture the estimate C_d(l,n) \approx nl/d and note that proving it would imply ES_d(l,n) \approx ES_d(nl/d), although this would still be far from the lower bound established in the paper.