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 ⊂ Rd, 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 ⊂ Rd with Bd ⊂ conv(Q), one can select at most 2d vertices whose convex hull contains r(d)Bd.

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.

References

The current working conjecture is that r(d) ≥ √α for some positive constant α.

— Quantitative Steinitz theorem and polarity  (2403.14761 - Ivanov, 2024) in Section 1 (Introduction)

The following question remains open.

\begin{conj} For every $d\ge1$ there is $\widetilde c_d>0$ with the following property. Let $X_1, \dots, X_{2d} \subset d$, and suppose that, for every $i \in \brackets{2d}$, there are $c_i\ind$ and an invertible operator $A_i$ such that $c_i+A_i\Bd \subset conv X_i, \qquad \abs{\det A_i} \ge 1$. Then there are a rainbow set $R$, a point $c \in C := conv\braces{c_i:i \in \brackets{2d}$, and an invertible operator $A$ such that $c+A\Bd \subset conv R, \qquad \abs{\det A} \ge \widetilde c_d.

\end{conj}

— A colorful quantitative Helly theorem for volume  (2609.25671 - Ivanov, 22 Sep 2026) in Section 4, subsection “Ellipsoids with different centers,” Conjecture 4.1 (labelled conject:different_centers)