On a conjecture of Grünbaum concerning partitions of convex sets

Abstract

In this paper the following is proved: let K ∈ ℝ² be a convex body and t ∈ [0, 1/4]. If the diameter of K is at least √37 times the minimum width, then there is a pair of orthogonal lines that partition K into four pieces of areas t, t, (1/2−t), (1/2−t) in clockwise order. Furthermore, if K is centrally symmetric, then we can replace the factor √37 by 3.