  We construct simple geometric operations on faces of the Cayley sum of two
polytopes. These operations can be thought of as convex geometric counterparts
of divided difference operators in Schubert calculus. We show that these
operations give a uniform construction of Knutson-Miller mitosis (in type A)
and (simplified) Fujita mitosis (in type C) on Kogan faces of Gelfand-Zetlin
polytopes.
