  Sublinear circuits are generalizations of the affine circuits in matroid
theory, and they arise as the convex-combinatorial core underlying constrained
non-negativity certificates of exponential sums and of polynomials based on the
arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of
sublinear circuits for polyhedral constraint sets.
  We give results on the relation between the sublinear circuits and their
supports and provide necessary as well as sufficient criteria for sublinear
circuits. Based on these characterizations, we provide some explicit results
and enumerations for two prominent polyhedral cases, namely the non-negative
orthant and the cube $[-1,1]^n$.
