Preimage Regions of Symmetric Separable Maps on the Simplex: Convexity and Barycentric Star-Shapedness

We study preimage regions on the open probability simplex associated with symmetric separable functionally generated maps. The problem is a finite-dimensional geometric question about convexity and barycentric star-shapedness of these regions. In the portfolio interpretation, the regions consist of the points whose generated portfolio has no negative coordinate. For symmetric separable generators, the defining first-order inequalities split into a coordinate term and a symmetric aggregation term. This coordinate--aggregation decomposition is the main organizing device of the paper. We show that the aggregation term may destroy convexity, and may even destroy barycentric star-shapedness. In particular, moving closer to the barycenter need not preserve the long-only property. We then give a necessary and sufficient threshold criterion for barycentric star-shapedness and derive sufficient conditions that recover it. These conditions are expressed in terms of concavity and second-derivative domination for the aggregation function. The entropy case is the affine aggregation case, in which the long-only constraints reduce to coordinate thresholds.