This study considers a simple newsvendor situation that consists of n retailers, all selling the same item with common purchasing costs and common selling prices. Groups of retailers might increase their expected joint profit by inventory centralization, which means that they make a joint order to satisfy total future demand. The resulting newsvendor games are shown to have non-empty cores in the literature. This study investigates convexity of newsvendor games. We focus our analysis on the class of newsvendor games with independent symmetric unimodal demand distributions after providing several examples outside this class that are not convex. Several interesting subclasses, containing convex games only, are identified. Additionally, we illustrate that these results cannot be extended to all games in this class.