@article{14660,
  abstract     = {The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set 𝑆⊂ℝ𝑑, then there are at most 2𝑑 points of 𝑆 whose convex hull contains the origin in the interior. Bárány, Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem. Let 𝑄 be a convex polytope in ℝ𝑑 containing the standard Euclidean unit ball 𝐁𝑑. Then there exist at most 2𝑑 vertices of 𝑄 whose convex hull 𝑄′ satisfies 𝑟𝐁𝑑⊂𝑄′ with 𝑟⩾𝑑−2𝑑. They conjectured that 𝑟⩾𝑐𝑑−1∕2 holds with a universal constant 𝑐>0. We prove 𝑟⩾15𝑑2, the first polynomial lower bound on 𝑟. Furthermore, we show that 𝑟 is not greater than 2/√𝑑.},
  author       = {Ivanov, Grigory and Naszódi, Márton},
  issn         = {1469-2120},
  journal      = {Bulletin of the London Mathematical Society},
  number       = {2},
  pages        = {796--802},
  publisher    = {London Mathematical Society},
  title        = {{Quantitative Steinitz theorem: A polynomial bound}},
  doi          = {10.1112/blms.12965},
  volume       = {56},
  year         = {2024},
}