  A classical result of Schubert calculus is an inductive description of
Schubert cycles using divided difference (or push-pull) operators in Chow
rings. We define convex geometric analogs of push-pull operators and describe
their applications to the theory of Newton-Okounkov convex bodies. Convex
geometric push-pull operators yield an inductive construction of
Newton-Okounkov polytopes of Bott-Samelson varieties. In particular, we
construct a Minkowski sum of Feigin-Fourier-Littelmann-Vinberg polytopes using
convex geometric push-pull operators in type A.
