The budget set is convex.
Remark 1.22
In establishing that the budget set is compact (exercise 1.231), we relied on the assumption that the choice was over n distinct commodities so that the consumption set is finite dimensional, X ⊂ Rn. In more general formulations involving intertemporal choice or uncertainty, it is not appropriate to assume that the consumption set is finite dimensional. Then, compactness of the budget set is more problematic. Note, however, that finite dimensionality is not required to establish that the budget set is convex (exercise 1.232).