A Problem Related to BÁrÁny--GrÜnbaum Conjecture

Pavle V. M. Blagojević, Aleksandra Dimitrijević Blagojević

n this paper we prove that for any absolute continuous Borel probability measure $\mu$ on the sphere $S^2$ and any $t\in [0,{1\over 4}]$ there exist four great semi-circles $\ell_1,\ldots,\ell_4$ emanating from a point $x\in S^2$ into four angular sectors $\sigma_1,\ldots,\sigma_4$, counter clockwise oriented, such that $\mu (\sigma_1)=\mu (\sigma_4)=t$, $\mu (\sigma_2)=\mu (\sigma_3)={1\over 4}-t$, and area$(\sigma_1)$=area$(\sigma_4)$, area$(\sigma_2)$=area$(\sigma_3)$