I am trying to solve an optimization problem and I am almost sure there exist some theorem which already does it: Let there be two probability density functions: $f_1(x_1,x_2...x_d),f_2(x_1,x_2...x_d)$
The d dimension space V can be is divided into 2, such that: $ V_1\cup V_2 = V $
The problem at hand is: Find the $V_1,V_2$ that maximizes:$\int_{V_1}f_1(x_1,x_2..x_d)$
under the constraint: $\int_{V_1}f_2(x_1,x_2..x_d)=C $