On convex complexity measures

Khrapchenko’s classical lower bound n2 on the formula size of the parity function f can be interpreted as designing a suitable measure of sub-rectangles of the combinatorial rectangle f−1(0)×f−1(1). Trying to generalize this approach we arrived at the concept of convex measures. We prove the negative result that convex measures are bounded by O(n2) and show that several measures considered for proving lower bounds on the formula size are convex. We also prove quadratic upper bounds on a class of measures that are not necessarily convex.