Weak extension of the ham-sandwich theorem : some ratio not in half

The ham-sandwich theorem is famous as an application of Borsuk's antipodal theorem. It states that given n measurable sets A_1,...,A_n with positive Lebesgue measure in R^n, it is possible to divide each one of them in half by a hyperplane. Recently the author applied Borsuk's antipodal theorem to optimization theory in [3]. In this paper we extend the ham-sandwich theorem to some ratio not in half by using the technique of [3].