We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

For a pair of posets $A \subseteq P$ and an order preserving map $\lambda : A \rightarrow \mathbb{R}$, the marked order polytope parametrizes the order preserving extensions of $\lambda$ to $P$. We show that the function counting integer-valued extensions is a piecewise polynomial in $\lambda$ and we prove a reciprocity statement in terms of order reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler [J. Combin. Theory Ser. A, 120 (2013), pp. 1372--1393] and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty [Notices Amer. Math. Soc., 54 (2007), pp. 708--717].