Abstract:
Khrapchenko's classical lower bound $n^2$ on the formula size of the parity function~$f$ can be interpreted as designing a suitable measure of subrectangles of the combinatorial rectangle $f^{-1}(0)\times f^{-1}(1)$. Trying to generalize this approach we arrived at the concept of \emph{convex measures}. We prove the negative result that convex measures are bounded by $O(n^2)$ 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.