We prove that for any k ≥ 4, any set X of points in the plane, and any point P ∈ interior conv(X), there is a subset Y ⊂ X of at most k points such that if conv(X) contains a disk with radius r around P, then conv(Y) contains a disk with radius [cos(2/(k + 1))π]/[cos(1/(k + 1))π]r around P. This generalizes the quantitative Steinitz theorem in the plane and proves a conjecture of Bárány and Heppes.
