Type: Article

On a conjecture of Grünbaum concerning partitions of convex sets

Arocha, J.; Jerónimo-Castro, J.; Montejano, L.; Roldán-Pensado, E.;

Abstract:

In this paper the authors prove the following result. Let K be a convex body (i.e. a compact convex set) in R2, such that the diameter diam(K) of K is at least 37???m(K), where m(K) is the minimal width of K (i.e. the minimal distance between two parallel supporting lines to K). Then, for every t?[0,1/4] there exists a pair of orthogonal lines that divide K into four pieces, with areas t, t, (1/2?t), (1/2?t) in clockwise order. The authors also prove that the constant 37??? can be replaced by 3 if K is assumed to be centrally symmetric. These results answer partially a question posed by Grünbaum, asking whether such a division is possible for every t?[0,1/4] and for every K, without further restriction on diameter and minimal width.