267 2019 2020 eng 171 195 24 article 1 2019-04-25 2020-06-01 -- We deal with a stationary model for flow through a network. The flows are determined by the values at the boundary nodes of the network that we call the loads of the network. In the applications , the feasible loads must satisfy some box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds. We present sufficient conditions to show, that the feasible set is star-shaped with respect to special points. For stronger conditions, we prove the convexity of the set of feasible loads. All the results are given for active and passive networks, i.e. networks with and without inner control. This analysis is motivated by the aim to use the spheric-radial decomposition for stochastic boundary data in this model. This paper can be used for simplifying the algorithmic use of the spheric-radial decomposition. Networks and Heterogeneous Media 10.3934/nhm.2020008 published Creative Commons - CC BY-ND - Namensnennung - Keine Bearbeitungen 4.0 International Martin Gugat Rüdiger Schultz Michael Schuster eng uncontrolled Validation of nominations, Stationary states, Isothermal Euler equations, Gas networks, Convexity, Star-Shapedness, Spheric-radial decomposition Friedrich-Alexander-Universität Erlangen-Nürnberg Universität Duisburg-Essen C03 https://opus4.kobv.de/opus4-trr154/files/267/ConvexityAndStarshapedness_Submitted.pdf