Quantitative Steinitz lower bound (order 1/sqrt(d))

Establish that there exists a universal constant c > 0 such that for all dimensions d, r(d) ≥ c/√d, where r(d) denotes the largest universal radius guaranteed by the quantitative Steinitz theorem; specifically: for every convex polytope Q ⊂ R^d containing the Euclidean unit ball B^d, there exist at most 2d vertices whose convex hull contains r(d) B^d.

Background

The classical Steinitz theorem states that if 0 lies in the interior of conv(S) for S ⊂ R^d, then there exist at most 2d points in S whose convex hull contains 0 in its interior. A quantitative version asks for a universal radius r(d) such that, for any convex polytope Q ⊂ R^d with B^d ⊂ conv(Q), one can select at most 2d vertices whose convex hull contains r(d)B^d.

Earlier work (Bárány–Katchalski–Pach, 1982) established r(d) ≥ d^{-2d}, and recent progress provided the first polynomial lower bound in [IN24]. The paper notes a prevailing expectation on the optimal order of r(d), suggested by matching examples that indicate r(d) cannot exceed a constant multiple of 1/√d.

The quoted sentence in the introduction explicitly labels this expectation as a conjecture, marking it as an open problem.