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.